Can I create my own parametric default character base model?

What if I wanted to make a city in Blender similar to Zootopia. Except I could not use a human generator when I need to use a custom default bunny, fox, goat, etc.

I’m wondering, what human generator would be the easiest to modify the base human into other animals?

I found some rigged animals using blendkit, including a dog, a cat, a horse, a bear, and some others. And they were very high quality.
(edit) also, I am not sure what you mean by parametric character. I understand parametric to refer to a type of optical illusion. If you are intending to make a city with a lot of windows, you might want to look into kitops add-on for making parametric room textures, which are kind of like holograms in a way. It takes only one polygon for an entire room interior.

Parametric refers to parameters to change a model. Such as Makehuman.

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To be honest maybe an incorrect assumption but the gist of it sounds like an anthropomorphic query?

If so, try DAZ or Makehuman for base mesh / anim / rigging etc…for either bipedal - quadrupedal conversion.

well you could write a python add-on or try get it into other programs but you would need a base mesh and morphs
what you can do is have morphs in the shape key options of the base mesh and use empties as drivers to create characters.