It may be possible, speak to the 2nd chargeholder about whether they would be willing to consider transferring it though I'm sure they want it set that it is only transferred if you go onto your parents' deeds on the same day.
It may be more persuasive if you could pay some of it off. They may want to know what you are selling it for (make sure it is for full market value or you're not selling it cheaper than you could) and what proceeds would be left over once the main mortgage is paid off and what, if anything, you would be paying to go on your parents deeds to see if it's feasible you could pay it from the proceeds.
They may charge a fee for transferring it and may instruct their own solicitors to make sure it's transferred properly for which they are likely to pass on the cost to you.
If I were them, I'd want to check the equity situation on the parents house ie their current mortgage debt (or anticipated if there a new mortgage is being put on in all your names) and make sure there was an appropriate restriction on the property to make sure that the property cannot be sold (or parties taken off the register) or remortgaged without their consent or them being paid off.
If you are going on your parents' deeds then their mortgage lender would have to consent to this - though if you are going on the mortgage as well then they may do a straight remortgage in all your names but make sure you are aware of any changes in rates and such if this happens, get a full illustration incase your parents lose don't end up on quite as good a deal.
You will also need to go through all the credit checks to go on their mortgage and there are likely to be fees involved though it may be possible to add these to the mortgage.
Their consent may also be needed to the second charge going on. They may charge a fee for this as well.
Hope this helps!