While I agree that bad pushups are better than no pushups until you fix them, your first paragraph is misinformation. Anatomy is a 2,000 year old science, not a personal opinion. As a point of fact, if someone flares out their elbows when doing pushups, this will cause two things. #1, it will increase recruitment of the chest muscles at the expense of the triceps, which isn't necessarily a problem. But #2, it will place stress on the shoulder joints, and over time, this can lead to shoulder injuries.
There is no good reason to do pushups in a way that will cause joint problems. If you want to build your chest with pushups, the answer is wide grip pushups, not flared elbows. Your parent may find they are quite good at wide grip pushups, this is common because the chest muscles are bigger and stronger than the triceps.
Given sufficient time, a trainer worth their salt will teach you to develop every muscle. They might very well have you do wide grip. But there are definitely incorrect exercises, those are the ones that harm you. Dead lifting with rounded shoulders is a classic example, lots of people ruin their back doing this. I do both standard and wide grip push-ups. Though personally to build the chest I prefer the bench and other forms of chest press.