Can I get reimbursed by Medicare if I hire a friend to take elderly parent to doctor's appointments, shopping, and light household work?

Your friend would have to be hired by your elderly parent, paid by elderly parent. Then at end of the year your elderly parent would have to report what was paid to the friend for tax purposes and your friend would have to claim the payments as income to the government and pay taxes on the income.
No medicare will not pay your friend.

No.

No, Medicare is a health insurance. It only covers intermittent care. If parent has a Medicare Advantage plan, it may cover something like this.

Am option would be calling Office of Aging and see if they have a Senior bus that can take parent to appts and shopping.