We propose a novel hybrid quantum computing strategy for parallel MCMC algorithms that generate multiple proposals at each step. This strategy makes the rate-limiting step within parallel MCMC amenable to quantum parallelization by using the Gumbel-max trick to turn the generalized accept-reject step into a discrete optimization problem. When combined with new insights from the parallel MCMC literature, such an approach allows us to embed target density evaluations within a well-known extension of Grover's quantum search algorithm. Letting P d e n o t e t h e n u m b e r o f p r o p o s a l s i n a s i n g l e M C M C i t e r a t i o n , t h e c o m b i n e d s t r a t e g y r e d u c e s t h e n u m b e r o f t a r g e t e v a l u a t i o n s r e q u i r e d f r o m 𝒪 ( P ) t o 𝒪 P 1 / 2 . In the following, we review the rudiments of quantum computing, quantum search and the Gumbel-max trick in order to elucidate their combination for as wide a readership as possible.