OP. I checked the IRS gas rate calculations, and 14*5 days a week would be 70 miles. With 56 cents per mile, gas costs would be about $39.00 and I truly feel now that in order for me to feel financially comfortable and secure, I would like $25/hr for the job itself, and parents reimburse for gas expenses separately. This would come out to $250 per week plus $40 for gas, so about $290 a week. I'm just not sure that they would be okay with me asking for almost 50% weekly increase. I asked to be paid on the books as well, and since they have never done that before, it would be an even higher cost for them.
I'm also dreading speaking to them because I met them through a sitter site and if they left me negative feedback over this (which they would have a right to), obviously it would reflect poorly on me. This is so stressful, but I know I need to suck it up as this is all on me for thinking in my head about the higher than area hourly rate and not the cost factors overall.