I could go on for ages, we tried to drive Route 1 but they had the worst rainfall in years and flooded the roads to save the bridges. So tip 1, always have a plan B and dress for whatever weather and crack on.
Also, depending what time you land, get your 4x4, ideally from the airport and do not go to reykjavik, do it at the end if you must- it'll suck up loads of time.
Here are some ideas on places... many depend on your parents fitness and weather but we found all the good stuff required a little effort - places where the tourist buses could not get.
- Go snorkeling in silfra (we drove straight here from the airport)
- Hike to the Reykjadalur hot springs
- Walk to the Sólheimasandur plane crash - I enjoyed this but it's a long flat and very boring walk.
Don't go to the blue lagoon - seek out other pools/springs. I've been told Sundholl is good in town but was being refurbed when we visited. There are plenty of other hot springs and pools in the golden circle area (and the entire country!).
There are endless waterfalls and things to see off Route 1 and in the Golden Circle but you might also want to consider going North instead. Though it sounds like you have already booked some hotels, so maybe an idea on area and we could name a few more places..
Also http://www.road.is/ check this website every few hours.