That looks more like a Ceanothus Silkmoth or Glover's Silkmoth caterpillar than a Cecropia Silkmoth caterpillar judging by the coloration. They feed on Ceanothus, but will also accept willow leaves as well as a bunch of other tree and shrub leaves.
Pretty much just feed it whatever it was eating. They tend to not like to switch host plants at this stage. It is in it's final instar, and will soon be spinning its cocoon.
Parasitoids are usually pretty specific. If they're not species or genus specific, they are usually at least order specific. There are exceptions, but I don't believe you have to worry about tachinid flies parasitizing mantises if they parasitize caterpillars.