I'm still on my "All The Adult Toons Basically Adopt The Warners" bullshit, so here's what I think they'd be like to their adoptive kiddos:
Bugs and Daffy: I love the concept of these two being like the rich, higher-up stars at Warner Bros. and I feel like because of that they'd be the Cool Parents.™ They would spoil the Warners with all their cool, big-shot toon privleges, and let fly around in private jets, have sleepovers at their mansions ect. I imagine if anybody ever questioned the Warners going in some place they would show up right behind them, flashing their I.D.s like "They're with us" and they'd just immediately have free reign.
Lola: I'm going with her characterization from The Looney Tunes Show here, just because I prefer it to the way she was in Space Jam. Lola radiates fun young mom energy to me. She'd probably do things like take the kids out to the mall and plan movie and game nights with them. She'd probably be the least responsible when it comes to serious parenting stuff, like oh the kids want to have ice cream for breakfast, lunch, and dinner? Alright, that's fine.
Slappy: We kind of already know how Slappy acts taking care of kids because of Skippy, and I feel like it would stay in that same vain of "Cool Sarcastic Wine Aunt." She'd probably be pretty protective of the kids too; It's on sight if anyone tries messing with them.
Pinky: Like Slappy, we already have an idea of what Pinky would be like with kids because of the episode with Romy. He'd absolutely dote on and smother them. He'd probably also be quicker than anyone else to defend them because in his eyes they are just precious angels who can do no wrong
Brain: Again we have the Romy episode for reference, so he'd be like the strict but still caring dad. Brain is also the "The Reason These Children Are Still Alive Parent." They all share a handful of braincells when it comes to the kids and he posseses all of them 90% of the time because "No, we are NOT letting them bungee jump off the water tower!" I think he would also be pretty proud of how smart the Warners are and impressed by what they've learned on their own (Yakko's Shakespeare knowledge, Yakko and Wakko's geography focused songs, Dot's interest in women's history, ect.)