A uniquely parsable grammar (UPG) introduced by Morita et al. (1997) is a kind of generative grammar, where parsing can be performed without backtracking. In this paper, we investigate a uniquely parallel parsable grammar (UPPG). We give a simple sufficient condition on morphism languages, and show that every such morphism language can be parsed efficiently in parallel by a UPPG. We show that the Fibonacci and Thue-Morse languages, which are instances of such languages, can be parsed in logarithmic time in parallel by UPPGs