A router, like MOST newer electronic devices, draws some power even when off; unless unplugged, (or on a power switched circuit that is turned off.)
There is NO advantage in turning it off for extremely short periods of time.
IF they have set the router security correctly to prevent unauthorized access then unless they also unplug it, they gain nothing.
Otherwise, if it is an "open" connection, (security NOT set,) turning it off prevents others from using it without their knowledge WHEN they are not using it.
(When they ARE using it, an "open" router could be used by others without their knowing it, so they SHOULD set the security to avoid this; especially if you have monthly limits on data transfer.)
However, it is their router, and you may just have to learn to live with it.
It should NOT take more than about 1 minute to reboot, and that should not be too long to wait to keep peace in the family.
I doubt you have anything to do on line THAT CRITICAL that it can't wait a minute or so.
Degree in Electronics, computer minor.
FCC licensed technician.