(Native American English speaker here.)
Actually, I think "behavior" might be OK, but here are the first two alternatives that came to mind:
The psychologists discussed how the Johnsons are raising their children.
The psychologists discussed how the Johnsons parent their children.
Yes, "parent" can be used as a verb.
If you really, really want a noun, and if you're not happy with "raising of" or "parenting of", here's what I got from a trip to Roget's Thesaurus:
The psychologists discussed the Johnsons' conduct toward their children.
The psychologists discussed the Johnsons' methods of child-rearing.
*The psychologists discussed the Johnsons' parentage of their children. [The meaning here is wrong, but I'm listing it for a reason I'll explain below.]
The psychologists discussed the Johnsons' influence on their children.
The psychologists discussed the Johnsons' effect on their children.
The psychologists discussed the Johnsons' management of their children.
The psychologists discussed the Johnsons' guardianship of their children.
The psychologists discussed the Johnsons' handling of their children.
Here's a phrasal verb, which can make only a clumsy noun:
The psychologists discussed how the Johnsons are bringing up their children.
These might be just what you're looking for:
The psychologists discussed how the Johnsons care for their children.
The psychologists discussed the Johnsons' care of their children.
Time to buy a thesaurus?
Since you now have quite a strong command of the English language, it might be time to buy a good thesaurus. Keeping a thesaurus in book form next to your computer can be wonderfully helpful. Unlike web sites, a book has no advertisements or flashing blinky-bloos to distract you.
Some people don't like Roget's original design for a thesaurus, which is organized by concept rather than alphabetically. However, it has its advantages. It places all the words and terminology for one concept in one article, and related concepts in nearby articles. For example, I found "parentage" in the (alphabetical) index, which pointed to an article listing a couple hundred words for reproduction. On the next page was an article listing a hundred or so words for ancestry, and then words for posterity, and then…"influence". Roget's thesaurus lets you wander around, sometimes fruitfully, sometimes leading you to unexpected surprises. "Care" turned up as a synonym of "treatment". Some web sites try to let you follow related concepts in a way that can't be done with paper, but the current batch tend to be big on animation and light on words.
Another good book to have is a dictionary of synonyms. Unlike a thesaurus, which is designed to help you search for a word when you already know the meaning you have in mind, a dictionary of synonyms explains the differences between synonymous words.