Gee, and here I thought that this movie was about alien abduction

The poster above me nailed it - she was molested by her teacher. The Parkers may not have been perfect parents, but they did not appear to be criminal, either. The cop put the molestation bug in Catherine's ear since it should always be considered in child disappearances. What I can't understand is where you are getting themes that are not there while completely ignoring the big themes that are right in front of our eyes.
In addition to being takin advantage of by her teacher, Lily also seemed to conflate sex with love, which made her even more vulnerable to sexual predators. A guy just had to be sweet and smile at her, and she responded. Her only "sexual" tie to her parents is her observations about her mother in her diary. She says Catherine is "just" a mother, and is unloved as a woman - as evidenced by the separate rooms and emotional distance between her mom and dad. Lily didn't see her mother as a complete woman because of this, and it resonated with Catherine when she read it since her mother also tied sex with love.
Influenced by her daughter's diary, Catherine tries to fix that "hole" in her life by offering herself to multiple men (she's seeking the love being withheld by her husband), crying "what's wrong with me" when they don't respond (she's not understanding that she can be loved for who she is, not what her body has to offer), and culminating in running out in the desert alone - which is what she thinks (at the time) her daughter has done.
Matthew, who refuses to sleep with Catherine and brings up her teenage promiscuity at the beginning of the movie, also has his wake-up call. He makes several remarks about not wanting his family humiliated in public and is furious with his kids' acting out to the point that he doesn't do anything when they slip out at night, yet he is confronted with the ultimate public embarrassment when Catherine walks naked through the street. For the first time in the movie, he reacts without anger - he covers her with his own shirt, brings her home and gently bathes her. Afterwards, they come together to lament their bad parenting decisions that led the family to this point. On the porch, Matt finally acknowledges his anger issues influenced his decision not to go after his kids - effectively abandoning them. Catherine leans into him, and it's the first time we see her show physical affection towards a man that is not rooted in sex; she finally is able to touch him with his needs in mind rather than her own.
In the end, they get their son back because his disappearance was tied only to his wandering at night. Their daughter, however, is likely lost to them since her disappearance was tied to her parents' being unable to see beyond their own personal issues. She didn't get the kind of parenting she needed when she was a teenager, and sought love from the worst possible place - an adult sexual predator.

Lily WAS molested. By her teacher. The rest of that? Didn't actually happen. At least, not in this movie.
They're coming to get you, Barbara!

WE HAVE A WINNER! No I had thought about this as well. Female Pedophiles are just fairly rare. And then there is the lesbian aspect of the abuse which less common. However, we can agree that abuse is occurring we just do not understand who is Perpetrating the act but the father's disdain toward his wife is also a clue. The more I think about the obvious scapegoat the father seems less likely but Tom's behavior suggest other wise. GREAT POST!

You're on to something, but you have picked entirely the wrong parent. The dad saw the kids leaving in the morning, he was watching them walk away through the window; what possible reason could he have for not just holding the information back from the police, but from his own wife; he was protecting the wife from the truth of what she had been doing to Lilly, abusing her. The mother is the one who molested Lilly, not the father. The dad was trying to protect his children and his wife, as he had finally figured out what has been going on, or perhaps he received that revelation when all the stuff with the school teacher went down prior to them moving to the new town. In either case, he was protecting his wife about the truth of what she been doing, but possibly had no recollection of.
And the wife was clearly had some very deep issues and showed all the signs of a sexual predator; her behaviour went waaayyy beyond a manifestation of sexual frustration, including the way she initiated sex with her husband. There are plenty of sexually frustrated woman that don't throw themselves at young boys and police officers, the whole concept of consent was completely foreign to her that's pathology in action. Maybe it's harder for us to see that because we don't think of women as sexual predators and molesters, but there are many, many out there.
The father went to the teacher's house, and the aboriginal boy's house desperately wanting to find Lily there because he didn't want to accept what he had been desperately trying to deny what was going on in his own household, with either him finally facing the truth for the first time, or that he thought or hoped the abuse had stopped. He knew all the places where Tom had been wandering off to, he had the places marked off on a map even!, so this was a common thing. The wife also was lying to the cop about Tom's wanderings, it was the father that interjected and said it was much more common that his wife was saying and he wandered much further than the wife wanted to admit to the cop herself. She was covering something up, and there's only one thing that she would be covering up that makes any sense.
When the cop accused the husband of molestation he was genuinely disgusted and offended at the very thought that he would do something like that; his reaction was defensive, but not for himself he was protecting his wife. Remember the one phrase from the girl's diary about the touching in the dark, "you touching me", that part. It was haunting the wife, to the point that she couldn't get it out of her head. She may have been what's referred to as "splitting" in psychology, she may have had no recollection of what she had done to her daughter, and possibly son, but with the consequence of what happened to her daughter because of her abusing her, the promiscuity, the disappearance, she couldn't escape the truth of the matter anymore no matter how hard she was trying to suppress it.
And that was the moment where she had a complete disassociative/psychotic break right before she did her Lady Godiva impersonation down Main Street, subconsciously, or perhaps even consciously if she finally admitted to herself what she had been doing, wanting to feel the shame that her act would bring in place of the shame she should have been feeling all along because of the abuse she was committing.
Think also of the scene where the husband said that Lily certainly didn't take after him. That had nothing to do with any promiscuity on his wife's part, that was after he finally realized what had been happening under his own roof, or perhaps already knew, but was trying to get his wife to admit it to herself. Whether he had just figured it out, was in partial denial, or that he had figured it out when the whole teacher-Lily sex thing exploded in their former town, that conversation was totally about his resentment of his wife for what she had done, and also trying to get her to admit what she had done. Abusers of children, especially ones that are acting our their own past abuse that could have happened to them when they were children, sometimes block out their own behavior. This isn't an unknown thing that happens, though it is rare, it isn't unheard of.
So many of the strange behaviours of both the father and the mother that were a little inexplicable, if you re-watch the movie with my scenario of what was going on under the surface, all those behaviors will make perfect sense.
And above all else, remember the conversation between the wife and the older aboriginal woman go to that part again, listen very carefully to what the old woman says to the wife, to the exact words she used. Somehow the old woman, I think the implication is through some sort of supernatural means, she knew what the woman had done, knew the denial she was in, her "splitting", and that she knew she couldn't help the woman because she was beyond any help. The wife knowing the truth of her actions would have destroyed her

My only point was that your ideas were crazy but you just got out-crazied and now seem relatively sane.

Debate your point please.

Crazy, just crazy.

Mrs. Parker raped her husband, attacked the cop and Birdie.
??? You see things that I didn't see happening.
I saw Mrs. Parker try to find something in the nature of comfort and solace in the arms of her husband, and for whatever reasons, he just reacts like a wet rag.
Then Mrs. Parker tries to seduce/entice the police officer and Birdie into having sex with her. That does not constitute a physical attack. Especially when she gives up when the police officer probably fears the consequences of getting sexually involved with someone involved in a police situation. And Birdie is probably cognizant that mother and daughters don't mix sexually and is still smart enough to not go down that path.
Unless the story offers some concrete facts otherwise, and it certainly didn't, people should go with charitable explanations, not with the worst.
If all this supposed molestation was going on, more should have been shown to let viewers know it is a real part of the story.

Lilly was molested. This seems to be an establish fact. The Cop mentioned to Mrs. Parker in the car. Mrs. Parker also accuses her husband of molestation. Tom is emotionally disturbed so there is also exposure to Lilly and her sexual escapades. Don't forget that Mrs. Parker raped her husband, attacked the cop and Birdie.

These aren't spoilers as much as an alternate script for a different movie