Hmmm... don't we already have petition zones of a sort? Offices are where petitions are already presented, no? Are there any petitions that happen right now in the game that do not involve going to an office? Perhaps the biggest improvement to petitions right now would be the book keeper or mayor or whoever to make records of petitions and these being (re)viewable.
That said, this discussion so far makes me want to see more depth to petitions, or, actually, social association between dwarves. I know this should be its own suggestion, but I think it would be cool if dwarves from within the fort were able to form social groups spontaneously on the basis of common demands or affinities. Individuals and these groups within the fort would be able to make petitions/demands of the powers that be in a fort, not just nobles and other individuals/groups that are hard coded to be able to right now. It could form the basis of some cool emergent gameplay where factions could form in ones own fort without it having to be based on some hard-coded things like guilds or religion (not to throw shade on guilds or religion in the game!). In a situation like that, I think dedicated petition zones could make sense, especially if there was more freedom for the player in how to structure society and political-related stuff in the game. In any case, though I think this should still be able to happen at meeting areas and offices as part of their function.
With regard to nurseries, I think they do make sense to have, especially if parenting-related labor and baby behavior is ever fleshed out more (which I am all for...). Have it function like a pasture zone. Even as things are now, it might be nice to have the choice of having a place to put children to help keep them from danger. In the interest of verisimilitude and fun, though, it would be ideal to have it where it would require the actions of caretakers to keep the children in the area, as children are prone to wandering off unsupervised and get into all sorts of trouble (in fact, tame animal behavior in pastures should be this way too).