I think you are already handling this pretty maturely for someone so young. It's one thing if you wanted to dye your hair blue before the family photo, but to tell you you can't wear it in its natural state - this sends a poor message. And I'm sorry for that.
Have you asked him why he feels so strongly about this? Or maybe the answer doesn't matter.
I think you could calmly say "I have thought about what you said, and though I respect your authority, I have chosen to wear my hair natural because I feel more comfortable. If you feel you still need to withhold my permit, that's of course your right to do so. Please understand I dont meant any disrespect, and I understand that you don't like how I look with curly hair, but it's what I have and I prefer not to change my appearance to please someone else - not even my father. I hope you will try to understand."
Of course put that in your own words! Lol!
Btw, you have beautiful curls, and I'm glad you are proud of your natural self!
If you do decide to straighten it, I would pull it back while damp into a tight twisted bun, and let it dry that way. (sleep overnight, even.). Then you can brush/blow dry the rest in the morning and it should be close to straight.
That's IF you decide to - which is totally your choice. No right or wrong answer here.
2c/3a/3b, Fine/med, Dense, high porous, & normal elas.
Lo-poo - Renpure Keratin, SMR, Giovanni SaS
clarifying - mix Johnson's H2T with low poo
Co-wash - Suave Nats. clar. or coco.
RO - GVPCB, Ion Eff. Care, TN.
LI - SMR, Cure Care, or RO
Seal - MO or Jojoba
DT - + honey, ACV, & EVOO to SM masque & GVPCB
PT - Gelatin!, CNPF
Styling - FSG, AIF, BRHG, LALS/PS, GF curl spray gel, SM Souffle/Smoothie, LOOB
Oils - MO!, EVOO, jojoba, GSO, SM Elixer, EVCO, VO5 Hairdressing