Hannah, yes you can take others flying all over Europe. All over the world in fact, as long as fly in a combination of PPL plus aircraft which is valid. But no, it will generally not be cheaper than taking a commercial aircraft, or a car, or any other form of transport. A PPL is to a very large extent a "fun" license, and only very few of us are able to get some serious practical use out of it - and this generally requires your own plane. (I'm sure IO540 will be along shortly...)
So to be honest, I cannot offer you any argument why your parents *should* pay your flying lessons. In fact, I don't think parents should pay for flying lessons at all, unless, arguably, you're aspiring to be a commercial pilot. But if you just want to fly for fun, get a job and pay for them yourself. That's how most of us did it. (But to be honest, most of the PPLs here started in their 30s or so.)