Yes I need to ask. A vacation is free if you don't pay for it. that's called free. I've done it many times. I agree with not asking for time off but I sure as heck wouldn't say to some one "instead of paying for my vacation can you just give me the cash?".
Even your own mother? I know that if I was in trouble, my parents would want to know and would want to help if they could. I can't imagine my parents even suggesting the vacation because they are very practical people and have taught me that needs come before wants. You don't take a struggling family member on vacation unless you are so fortunate that you are already able to meet that person's needs and can afford their wants.