What kind of benefits when it comes to elder care does a retired teachers on a pension have?

They have the same "benefits" and options as anyone else. If they have long-term care insurance they have any benefits spelled out in the policy. If they have money or assets they can spend almost all of it and when they run out they'll be eligible for Medicaid. Medicaid does cover in-home care.

In no other developed country do people go bankrupt paying for medical care. The US is the only one with that distinction. It is shameful.

If your parent is eligible for Medicaid, don't let pride keep you from applying for her. We might think of it as for people in abject poverty but it really for anyone who can't provide their own medical care because of age and/or disability.