Does NM state provide financial assistance to people who take care of their elderly parents in their own home?

Vast majority of grown children do not get paid for taking care of their parents, unless the parents pays them. If your parents can do that, make sure your draw up an employment agreement.

What is in Colorado is a "Cash and Counseling" program. Each State is different with their own programs. Some have "Cash and Counseling" via Medicaid, some States have said program not associated with Medicaid. You'd need to contact the New Mexico social services to see what is available. If Medicaid funded then your parents would need to apply and qualify.

As for a salary it varies from State to State and depending on the current State budgets. Probably would be less than a part-time job at minimum wage.