Smoke rises and goes through those fine even tho their narrow. But rain almost never falls perfectly straight down, so the amount of rain that gets in those three narrow pipes is incredibly minimal.
If the fireplace was still open with the tubes it wouldn’t be an issue what little water might come in would just evaporate.
It’s probably ok. But you usually see that method in Europe or upper east coast America in historic homes.
You’re probably better off asking a neighbor, or if you’re really worried, just call a chimney inspector so you know for sure