We consider the synthesis of distributed implementations for specifications in parameterized temporal logics such as PROMPT-LTL, which extends LTL by temporal operators equipped with parameters that bound their scope. For single process synthesis it is well-established that such parametric extensions do not increase worst-case complexities. For synchronous distributed systems we show that, despite being more powerful, the realizability problem for PROMPT-LTL is not harder than its LTL counterpart. For asynchronous systems we have to express scheduling assumptions and therefore consider an assume-guarantee synthesis problem. As asynchronous distributed synthesis is already undecidable for LTL, we give a semi-decision procedure for the PROMPT-LTL assume-guarantee synthesis problem based on bounded synthesis. Finally, we show that our results extend to the stronger logics PLTL and PLDL.