Not static browser ones. Closest game with wait/cooldown times I can think of is Furvilla that has building, minipet breeding, and other varying mechanics but even then it differs vastly from Pixpets.

Edit: Honestly if you want you can also try out Dragcave but... that's literally just a collection site. There's no mechanics aside from breeding your dragons and collecting them so unless your thing is endlessly hoarding dragons and maybe focusing on lineages in the process go for it. There are cooldown times there too but no explore mechanics or growing or anything. Just hoarding pets.

Honestly though you can look at Pokemon sites for the sprite aspect or Phone/facebook games for the collect+wait based aspect (if you have access to those)! Some have little explore mechanics, many have breeding and growing mechanics if you know where to look (most pet games are literally exactly the same), and each one generally has a few unique stuff that makes them special. (I've explored a lot around the realm of phone games and I think the one I stuck with the longest is Dragonvale which is also accessable via facebook if you actually brave being on facebook but there are many others that are similar to it)