@Nick Cooper,
Everyone else's comments are correct. There's really not much benefit financially to paying cash and then refinancing. It very well may end up costing you more.
But, here's a question no one else has asked... What if you can't refinance the way you think you can? If you go to refinance and there is no loan in place, it's a cash-out refi. The rules are very different in those scenarios. There are seasoning requirements, liquid asset requirements, cash flow requirements, credit score requirements, etc.
Here's a few questions about your parent... What if something in your life changes before you get to refi? How will that affect your parents? How long are your parents willing to lend you the money? Do they need it back by a certain time? What if you don't meet that timeline?
Borrowing from family & friends is great. But few actually think about the consequences of what will happen if/when something goes wrong. I've yet to meet an investor who will answer more than 0 to the following question...
How many investors (including yourself) do you know of or have heard of that have gone out, negotiated what they believe to be the deal of a lifetime, get it under contract, escrow open, deposits made, contractors lined up and chomping at the bit to get started and drool running down their chins who went home to tell their significant other that they can't wait to lose a ton of money in that deal?
The question is a very real question (albeit a bit sarcastic). Borrowers/investors rarely ever ask themselves any questions remotely close to this. Why, because every single one of us thinks that every deal we want to by is a good deal. That's why we want to buy it. Except, things happen. Deals go bad. Deals have hiccups. Unexpected things happen all the time.
So... the real question is... Do your parents understand the real risks or are they just offering to help because you are their kid?
Before you make a decision as to which way to go, you should have a deep meaningful financial discussion with them about the risks. If they still agree to do the deal with you, then go for whatever works best financially for both you and them.
Good luck!