It's grouped as:
One of [the thick tubes that carry blood from the heart to other parts of the body].
So "carry" agrees with "tubes".
[One of the thick tubes] that carries blood from the heart to other parts of the body. is, I suppose, theoretically possible, and would mean there are many thick tubes having different purposes, and we're referring to one of those whose function happens to be as stated. If I meant this, though, I would rewrite it to avoid possibly ambiguity or misinterpretation.