Gerontocracy is a noun that refers to a government ruled by the elderly. 

The Greek word geron (Jer ON) means ‘old man,’ while the suffix C-R-A-C-Y — also seen in words like democracy, plutocracy and meritocracy — refers to ‘power’ or ‘rule.’ Gerontocracy has been around since the mid19th century. Here’s an example:

With all those city council members in their sixties and seventies, I’m beginning to think our city is run by a gerontocracy. I have nothing against the elderly, I’m just worried that I’ll have to wait too long before it’s my turn to have any power.