OP, there are so many unanswered questions that it makes it very difficult to judge whether you or they ABU. I suspect a little of both.
Do they each own their £1.75m-ish homes, or are they mortgaged?
Do they have partners/spouses and any further children?
Did you phone rather than a face to face conversation?
Did you request a loan/equity release that you had researched, rather than asking whether they could offer any help or advice?
How much is the home you are looking to buy, and how much mortgage could you secure? £200k as a deposit on a £300k house is a 60% deposit so will give you access to great mortgage deals, but if you are paying your own £100k mortgage at, say, £500pcm then you would be looking to pay your parents' loans at 4.4% x 200k so around £700pcm (interest only) so that's £1200 gone before you can put food on the table.
Is your DC father paying maintenance? He is obliged to contribute towards housing you and DC.
Did you suggest buying the house to your parents as an investment if they had capital free, and renting it from them to make it fairer than them gifting you money from the inheritance you might or might not get eventually?
Obviously we're not asking you to post your finances on here, but unless you are earning around £25k you would be unlikely to get a £100k mortgage, but if you are only earning £25k you would not have enough take home to support the mortgage payment, equity release payments and live on (unless receiving decent maintenance payments).
I wish you all the very best of luck in finding a lovely new home for you and your DC 