A picture would certainly help but it sounds like you have what is called a sport. A sport is a naturally occurring genetic mutation. For what ever reason, while the plant is growing the genes get mixed up differently and a different flower appears. It can be a different color, shape or size bloom. If a cutting from that section is rooted and grown and the blooms continue to be different from the mother plant it's called a stable sport. There are many examples of stable sports that are in commerce today. Many "climbing" forms of hybrid teas are sports of the original bush forms. You will see names that say "white' this or "red" that and often (not always) these are different colored sports of the original rose.

If you'd like you can try and root a cutting and see what happens. Or you can tag that branch and see if it continues to bloom differently.

Enjoy!

This is so interesting - thank you for responding! I am far from a gardening expert much less an expert on roses. I was learning about grafting but from what I can tell, if that happened I probably would have seen the separate blooms long before now. I hadn't found the word "sport" yet so will investigate that now. I'll try to post pictures tomorrow if for no other reason than it's a really beautiful plant. Thank you again for taking the time to explain!

Could be a sport (mutation). Also could be a sucker from the rootstock. Where does the cane that the pink flowers are on emerge? If they are coming from below ground, then it could be a sucker from a graft. Do the blooms looks like the link below? Then your blooms are the rose Manetti, which is suckering.

Here is a link that might be useful: Manetti at helpmefind.com