Hello, Congratulations to you and family. What I did before, 5-6 years ago and I can confirm it is still happening, just tell you parents to print the letter out, take their photos and passport and printed letter to a travel agency in Iran. They will charge them a fee and will take their passport and documents to Turkey and will get it processed in a VAC office and as soon as they get it back, they will bring you the passport and COPR back.
other option is for your Mother and brother to travel to Turkey themselves and go to VAC and get it processed themselves. And because the embassy is in Ankara, of course going to VAC office in Ankara will be much faster.
good luck and all the best.
Just curious! If the parents go personally to VFS Turkey, Would VFS Turkey return the passport on the same day or how much time do they take? Because I am assuming the parents will need their passports for returning to Iran.