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.