Well, I'd say there is something in common between a mother and a father when it comes to what they do to their children, and I can't think of another word than "parenting" that would cover it.

The two requirements for a new word is that it should be necessary/useful, and that it should sound right. The latter part is where most of those words ending up "ize" fall down. Often enough - bu no means always - they are meeting a need for a word, but they just do it so badly. Which means the vacancy is still open.

But very often, if you hunt around, the word you need is there all the time.