I'm confused as to why you're using different solicitors. There's no conflict of interest, so you could have saved money by using the same solicitor. (When I sold part of my garden to a neighbour we did exactly that).
Indeed, since there's no money involved, I find it hard to see why you would need a solicitor at all. All that's required is for your parents' solicitor to ensure that this form is properly completed and to submit it to the Land Registry:
https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/449594/TP1.pdf
So your own solicitor hasn't got anything to do!
However, if you've engaged the services of a solicitor anyway, he/she will not unreasonably expect that you must want him/her to do
something (such as carrying out the searches that are normally involved when property is transferred for a consideration), so it's not unreasonable that he/she would want paying for it.
You need to ask yourself why you've employed a solicitor. If you can't think of any good reason (and I certainly can't) then dispense with his/her services but expect to be billed for any work already done.