Boot an older iMac as a HD on new 2013 iMac

The Target Disk mode is triggered by holding down T right after the startup sound (I think).
But as far as I know it only works via Firewire, so you would need a Thunderbolt to Firewire adapter.

Alternatively you could remove the old HD and use it as an external HD.
May I ask what model/year it is from? iFixit probably has some guide for removing/switching old HDs for your Dad's iMac

 

It's possible that the drive in the older iMac is still ok (hardware-wise), but just has some kind of software problem that keeps the iMac from booting. Then again, it could be an actual hardware problem (i.e., failing drive mechanism).

In either case, prying open the iMac might be more trouble than it's worth.

My suggestion is to spend $30 and buy the thunderbolt-to-firewire adapter. Then try a standard "target disk mode" connection to the older iMac.

It will cost you the thirty bucks, but if you discover the drive still won't mount (hardware failure being the most likely cause), it will have saved the time and trouble of opening the iMac, ending up with a table full of parts, and still not be able to "get at" the data on the drive.