I got my first parrot, Birdie the jenday, about a year after my first birds, which was a pair of ringneck doves. They never got along, I never even really tried to socialize them together, it would be dangerous and pointless... I think your question would be more regarding when I got my 2nd parrot species that DID flock with my first conure.
My 2nd parrot was a cockatiel named Kibble and I didn't even really mean to expand my flock, my then-roommate moved out of the country and asked if she could leave Kibble with me instead of with a shelter and I was so happy. That was maybe a year after I got Birdie. It happened very organically. Actually, while I've adopted quite a few birds since then, I didn't decide on my own to adopt another bird until like 5+ years later.. I just kept acquiring birds from friends lol.