The third and fourth generations, each with their own Robins, appear as babies and toddlers. That leaves plenty of opportunity for the older kids to babysit. But not really all the kids, at least equally.
I think the babysitting theme started in issue #4, which DC marketed with this line:
Meet the Little Tiny Titans as they show Wonder Girl just how tough babysitting can be!Inside, Wonder Girl and another heroine named Bumblebee babysit four teeny-Tiny Titans while Robin, Speedy, and Cyborg go play baseball.
The copy for #15:
And it’s Rose's turn to babysit the Terror Titans when a group activity yields fiery results.In #23, Robin visits Batgirl and meets two of his little fans.
When Raven has to watch Kid Devil for the weekend,…Have you noticed a pattern?
The cover at top is #29, which came with this copy:
It's Supergirl's turn to toddler-sit the tiny Tiny Titans! Can she handle this crisis of infinite toddlers – or will she burst into tears?
The story in issue #26 has Beast Boy watching over Miss Martian, but the word “babysit” wasn’t part of its marketing copy, and Beast Boy actually insists it’s not his turn. Even if we agree that he really is babysitting here, he’s the only example of male parity I could find. (Did I miss an important panel?)
Most recently, Tiny Titans, #47:
It’s Mrs. Atom’s turn to babysit the Baby Titans – Damien [sic], Arthur Jr., Smidgen, Kid Devil and Jason Toddler. Can she do it alone? Maybe Miss Martian can help!And Bumblebee is back to babysitting as well. Even when a Tiny Titans story involves multiple baby-sitters, all three are female. Furthermore, Mrs. Atom is a character with no equivalent in the regular DC Universe. [Trust me, Ray Palmer’s love interest Jean Loring is not the same.] She seems to have been created to care for her children and others.