I believe there's a general statute against harassing wildlife in NM, but I couldn't find a good reference.
There are pretty strict requirements for using dogs to chase animals in NM as listed in the rule book here. While it doesn't specifically state that you can't let your dog chase wildlife, it provides enough specific conditions that one could infer chasing (an activity not specified) is not allowed.
It would definitely be illegal on NM state land:
Originally Posted By: nmstatelands.orgProhibited Activities:
1) disturbing, harassing, injuring, destroying or removing wildlife, livestock, or any other animal life;
It appears to be illegal on BLM land as well based on this:
Originally Posted By: Watchable Wildlife page on BLM's NM page•Never chase or harass wildlife; harassment of wildlife is against the law
In Grant county we have specific leash laws. The only time a dog can be used to chase game animals is when a licensed hunter is with the pursuit animals. Dogs cannot be used for deer and elk even for blood trailing. Technically your dog should be on a leash even when hiking. I have people all the time complain about there dogs getting caught in traps and the first thing I ask them is if their dog was on a leash. They always say no. So I give them the talk about not complaining about a legal activity when they are the ones breaking the law. I understand letting your dogs off a leash when you are up in the hills but you should be able to control your dog even if unleashed. If you can't don't turn them loose, or don't complain if they don't come home.
No specific knowledge concerning NM but think your father doesn't know what he is talking about. If the law is the same as here he will wind up with a dead dog and a very large fine. Both he can live without.
Check with the head game wardens office and request an answer in writing.