What does California housing require as a reasonable, not excessive pet deposit for a landlord to charge to tenant ? Is this supposed to be only a one time up front deposit?
Do some landlords in mobile space parks have exemptions and can charge this monthly?

There is no state law regarding that issue. Its on a case by case basis. Imagine you want to rent and have 3 cats and 2 dogs. Your deposit may be different then a person who has only 1 cat or 1 dog. Also many laws governing rentals are handled on a local level. You do have renters rights. But as far as deposits that can vary widely.

In California wouldit be legal in a mobilehome park to charge a fee or deposit every month of tenancy for 1 or 2 dogs, or only a
First time move-in deposit or fee.
I realize deposits may varydepending on ho many petsare in the mobilehome tenancy.