Hindu deities mount and travel on animals which are their vahana. They are always depicted with them. We must therefore respect all these animals too.

Note:
Hanuman, the monkey god, does not have a vahana.

The depiction of a Hindu deity using an animal or mythical entity as a vahana should obviously not be a reason to justify using animals for the purpose of travelling. The depiction itself may be symbolic rather than factual with the animals being ridden representing the qualities that the particular deity has mastered. For example, Goddess Durga mastered anger & ferociousness, Lord Ganesha mastered stealth, Lord Kartikya mastered pride and arrogance, Goddess Parvati and Lord Shiva mastered indolence, Lord Bhairava mastered death, and so on.