It's great that you want to communicate respectfully with your friend's mom. However, it's also important to consider her perspective as a parent. Instead of asking her to let you continue speaking with her daughter, you could approach the situation with understanding.
Here’s a suggestion for how you might frame your message:
"Hi [Friend's Mom's Name],
I hope you’re doing well. I wanted to reach out because I respect your concerns about [Friend's Name] communicating with friends. I understand that you want to make sure she is safe and surrounded by positive influences. I really value my friendship with [Friend's Name] and have always tried to be a good friend to her. If it would help, I’d be happy to discuss the nature of our friendship and reassure you about my intentions.
Thank you for considering this, and I respect whatever decision you make.
Best,
[Your Name]"
Remember to be polite and respectful, and keep in mind that her mom may have her daughter's best interests at heart. Good luck!