i never paid much attention to the parasite issue... but looking at this closely for about a week, i do suspect that Max may have picked up some type of skin parasite that looks very much like eczema. which one came first, who knows... but, you might consider checking into internal and external parasites.
for the skin mites, pure neem oil would be a good rec. or bathing in water with some tea trea oil... neem soap.
I thought parasites were irrelevant, but now i plan on leaving no stone unturned. hang in there, and I wish you the best.