The conditions for the chimp to human jump require a series of changes where each brain increase enables better language/tools that pays for the increased costs.

Parrots/birds don't seem to have a feasible path like that - light bodies designed for flight, lack of hands. Cetaceans can easily grow and support large brains but fire doesnt work under water and most tool potentials are limited. Elephants seem to be the most likely runner up, if primates weren't around - perhaps in a few tens of millions or hundreds of years there could have been a pachyderm civilization.

So yeah - it might be somewhat rare, but its hard to say, as it didn't take that long on earth.