Kind of. Mutation rate of our dna is "managed" by the dna/chromosomes/genes to reduce the rate in critical areas.
> the chips are stacked against them.
Wikipedia says: "This reproductive method enables the asexual desert grassland whiptail lizard to have a genetic diversity previously thought to have been unique to sexually reproductive species."
Doesn't look to bad?
I'm not sure which Wikipedia article that is or if there has been revision between when you checked, I didn't find that in the Wikipedia articles for Telidae or Parthenogenesis. The Parthenogenesis article notes that it's controversial whether this is a threat to their ability to adapt.
So I may have been wrong with that example and I thank you for correcting me. I stand by the statement that evolution can get stuck in local minima but I may have been wrong about the Telidae.