A purist will tell you that all the elements should agree. Plural parents have plural children. Therefore, they say, it should be “parents pick up their children,” not “child.” But the solution creates some problems of its own. For one thing, it could be construed to mean that every single parent involved has “children” and that, therefore, no parent can have just one. It could also be construed to mean that the children are “their” children, with the “their” referring to all the parents — one big happy family, though probably not what the writer intended.
Words like “each” and “respective” can get you out of some pickles. But they get old pretty fast, especially when all your wording reflects an iron-willed commitment to precision: “Each parent should pick up his or her respective child or children.”
So, what to do? The truth is, no one really knows. But Atlantic Monthly columnist and “Word Court” author Barbara Wallraff offers some sound advice: “When one is at pains to make clear that the individuals in the subject are to be paired one apiece with the persons, places or things in question, the number of the noun can’t be relied on to make the point, and other clues must be given. (‘Each of the bigamists relied heavily on his wives.’)”