Two teeth extractions to continue the Orthodontic treatment? (photo)

I am currently having braces for five months and my orthodontist said me that's all she could pull back my teeth without extraction. She suggested me to extract 5/5 from my upper jaw and continue the treatment or otherwise she would start the finishing phase. What should I do? I am 24 and my parents strongly disagree for extracting teeth. I have attached photos of current situations of my teeth herewith. Please give me an advice. What's the best thing to do?