Can only 1 parent(green card holder) apply for minor child to immigrate to USA from foreign country ? Parents are Married, only one is in USA.

I am a US citizen, my parents live in Europe, They have a minor child (10 years old) which is my sister. I really want to get my little sister to eventually move to USA. I know that petition for sister of US citizen takes 10+ years, so i was thinking about different approach to it. I wanted to file a petition for only 1 of my parents(mother) to immigrate to US and get a green card, and as soon as she gets it, she would file a petition for unmarried minor child to immigrate to USA. Question is, is this possible ? since my parents are normally married and both have joint custody of my minor sister. If it is possible, does mom only need Dad's permission and signature to file a pettition for their minor child to immigrate to USA. Going this method, by processing times calculation, it would take only 2-2.5 years to get my little sister to USA. Both of parents are onboard with this, and mom would eventually move back to homecountry in Europe. Thanks upfront for replies and suggestions.