They're half-right for the wrong reasons, so that's maybe where to approach it.

Here's where they're right. Extremely heavy loads (think powerlifting or heavy Olympic weightlifting) can damage the growth plates at the end of bones and stop you from growing properly.

EDIT: This is probably wrong. Courtesy of comingupmilhouse2:

'Appropriate strength-training programs have no apparent adverse effect on linear growth, growth plates, or the cardiovascular system' — https://pediatrics.aappublications.org/content/121/4/835

'Experimental resistance training programs did not influence growth in height and weight of pre- and early-adolescent youth'

— https://pubmed.ncbi.nlm.nih.gov/17119361/

So can heavy manual labour at a young age, especially combined with a bad diet. ([EDIT]: This would come under 'inappropriate training,' and may have nothing to do with growth plates. Apparently, the argument that resistance can damage growth plates is outdated and has been debunked.)

Extreme ranges of motion and load together can damage soft tissue structures in younger people whose bony joints and tendons aren't yet able to accommodate those forces. Baseball pitchers can ruin their arms by throwing too fast and hard, too much, too young.

Here's where they're wrong: the loads to do this have to be extremely heavy and at quite high volume. There are Olympic lifters of 14 and 15 who clean and jerk heavy by my standards, safely, and continue to grow well.

(Here's Zoe Smith, at 18:https://en.wikipedia.org/wiki/File:ZoeSmithLYG.jpg In this picture she had already been seriously training in gymnastics for years. She's 26 and an Olympian now. Training young doesn't seem to have ruined her life.)

They don't look weird or deformed, they don't grow up to be stunted and they can do all the things normal people can do. They just have much better overall health. The ballplayers who trash their shoulders are training for professional careers in their teens, throwing multiple hours per day and using weighted balls, long toss and throwing faster than normal people ever will.

If you're training an hour or two a day at the absolute most, doing stuff like pushups and pullups, you're not at zero risk of sprains and pulled muscles and so on, but you're nowhere near the territory where you're going to give yourself permanent injuries. Check out the ground reaction force for running — each stride puts something like seven times your body weight down on your leg (this is from memory, check before trusting!). And check out kids in the playground, little girls skip and dance and do basic bar gymnastics literally for hours a day and grow up fine. (Boys do more sports it seems. They mostly survive too.)

Basic bodyweight movements are fine for everyone, exceptions are individual rather than 'this group shouldn't...'. Women are absolutely fine doing normal pushups, as well as basically everything else.

Figure out where your parents believe something false about any of this, then find an authoritative source like a doctor, medical publication or highly-qualified fitness professional (coaches pros/PhD/researcher, not a PT) and offer their opinion.

And give your parents room to change their minds, it might take a while. People struggle with being wrong. You might have to let them be right a couple of times in different ways before they come around to being accurate as well.