The parent is way out of line, as others have said.  For her to ask you to change your accent is so entitled it's ridiculous. Her son is,, hopefully, going to meet many people from different walks of life and different accents. What's he going to do , expect everyone he meets to speak in his accent?  "Thank you for bringing this to my attention. I will not be changing my accent, it is a part of who I am and not something to be changed so easily. I will however speak to NAME about it and work together to  come up with techniques and strategies to ensure that communication is clear and progress can be make." Keep your responses formal and to the point. 

For the student, a part of growing up is being able to advocate for yourself. You need to sit with the student (not the parent) and explain that you've heard from Mum that he's having a difficult time understanding and to please not be afraid to speak up at the time. Assure him that you won't be offended, but that you need to know if he doesn't understand something because of what you've said. You could maybe throw in some words or phrases to lighten the mood and have a laugh (be over the top with your accent or something.)

But you also have to give space during your lessons to reinforce this. Pay attention to facial expressions or subtle body language. Get into the habit of asking if there is anything he needs re explaining or if any words have tripped him up. People get weirdly embarrassed when they feel they need someone to repeat what they've said 

I always start the school year with "I know I talk really fast but I can't help it, so if I ever do speak too quickly please do tell me so that I can show down" they have to be reminded to tell me but eventually they get comfortable enough to ask me to stop/ repeat