First, do you need a car? If so, what are all the things you need it for? Will the 2006 Prius meet those needs well? Assuming it is something you actually need that meets your requirements...
Look at tire tread life, if it is on the second set of tires they are probably ready to replace, there may be lots of life left if it is the third set. Expect to pay $400-500 for a good set of four tires installed, be frugal but not cheap with tires. I'd also consider exhaust system condition if you are in an area with snow and salty roads. That is a predictable major expense if the system is nearly rusted through. We decided to trade in our Gen1 Prius when it came time for a new exhaust at 110k miles. The exhaust, only available from Toyota, was $1800 before installation, no aftermarket available. These things have elaborate emissions control systems. It also needed tires and had the original gen1 battery.
Will your parents be cutting you a deal? When I was younger I bought my dad's old truck when he replaced it. He secretly fixed a few things on it before it came to me and he told me to name a price. He accepted my initial offer. I think he would have accepted a bag of oranges.
If you have no idea how to approach price with them, I would be honest that you don't know what to offer and give them the private sale blue book values and any anticipated maintenance costs, then hear their thoughts. Obviously lower is better, but you don't want to take advantage of them and have any hard feelings. But if they want to help you out by offering a slightly low price, that could be ok, depending on the family dynamic. If this fosters dependence or they will lord it over you, don't look for a break, maybe even buy a different car. If they would otherwise just take it to a dealer for trade, offering to buy it from them at the same price the dealer offers will probably be lower than $6k and a good deal.