Does using animal models for medical research also promote veterinary research on these animal types?
Short answer is no. Human disease take a long time to develop naturally, far longer than the lifespan of animals models used (mice, rats or fish). 

Also the cost of the drugs far exceed the cost of the animal. There is little point in spending $40k to make a non-commercial anti cancer drug to save the life of a pet mouse who has developed the rare cancer. 

Longer answer... some... a bit. Not quite what you expect. There are a few labs that are re purposing anti cancer drugs that failed animal (mouse) trials to treat dogs. It appears that people are willing to spend a fair amount on experimental drugs to save the life of the family dog. 

Mice have increasingly been shown to be a poor model for human anti cancer drugs. Their cells grow to fast, their immune system is subtly different from humans. Stuff that work in mice often times are too toxic in human or just don't work. A longer lived animal model is needed. However dogs as experimental animals is too costly to maintain, too much red tape to use and takes too long for cancer progression (You cannot wait 2 years for a cancer experiment to finish). 

However a sick family dog with cancer is another matter.