Two objections, at the very least:
Running (many) chains in parallel reflects on the distribution of the starting values as we cannot be sure to "reach" stationarity for all of them in a finite number of steps. Hence a bias.
Weighting MCMC values by their likelihood means the likelihood is counted twice (as a power of two!), since the values are approximately distributed from the posterior, i.e., the prior x the likelihood. Hence another bias.
Now importance sampling may be associated with MCMC, as we recently demonstrated.