You don't have a vulnerable parent in their 80's that hasn't been vaccinated yet I guess?
If you have that situation, it sounds like the problem is with the GP. All 4 parents in our family (in their 70's) have received their first jabs - they don't have specific vulnerabilities. A friend's sister is a GP and they have completed the 80's cohort in that practice and are working through the 70's. I know it can be a bit patchy but sounds like your parent's GP is behind the curve a bit