my mom won't let me swim while on my period?

I'm 15.

my mom and my friend's mom are also friends. So tomorrow, my mom, me, and my brothers are going to the park, and there's a swimming pool there. We're going with my friend, her mom, and my friend's brothers and sisters. So anyways, my mom is friends with her mom, I'm friends with her, our brothers and sisters are friends with each other, etc.

They were over yesterday and her mom told my mom that we should go swimming tuesday, since both our older brothers don't have work tomorrow. I told her I had my period, so she insisted we just don't go to the swimming pool, but our older brothers should. I felt bad and I told everyone that they can swim and I don't care, I'll just stay at home. My friend rejected and said she doesn't wanna swim alone, so we'll just not swim. Our moms believe that it's dangerous to swim on periods. I told my mom it's perfectly safe, and our health teacher taught us, I read articles online etc. I told her to even call my doctor and ask him. She started saying, "well still, it's disgusting", and I'm like, well when you're swimming your flow stops or becomes really light, so there's no blood to be disgusted of. (I can't wear a tampon or cup, I'm not allowed to open my hymen). She then said to just stop the conversation and that I'm not swimming if I'm on my period.

Is there anything I can do or say to convince her?

Even with facts she's resisting, but instead decides to stick to the fake tales she learned while growing up.

Thanks