Could such Sociological Research happen?
Given that there's a dearth of reliable scientific research into transgender children, and given there are more pressing topics (e.g. efficacy of different treatments that are already occurring), I doubt it would be a priority.
Plus, it would need to include trans-identifying kids and non-trans-identifying kids and be quite a huge (expensive) study. Also, you'd really need to start it from pregnancy and follow up at various points to avoid recollection biases when parents are trying to remember what they did several years ago, not least for the ones who are now very conscious of their trans-identifying child's narrative around sex stereotypes that they will have repeated to doctors and everyone else over and over, or for the parents who now make a conscious effort to eschew stereotypes.
Furthermore, the question would need to be narrowed significantly to be useful - while a gender reveal party might signify the parents have strong views on adhering to sex stereotypes, there are lots of other reasons that parents might have this kind of party (because all their friends did, just to have a party, for the insta pics, to get gifts, because they're excited to share their baby news, and so on).
I think it might be better to come up with a questionnaire that reveals the parent's beliefs around adherence to sex stereotypes generally as well as parenting questions, and make inferences from that.
I think in many instances, imposing strict adherence to sex stereotypes in childhood often does affect a person's understanding of who they are in the world and can foster an uneasy sense of being at odds with society's expectation of them. I've heard lots of middle-aged transwomen describe how they knew they were a girl when they were told off for playing with "girls'" toys in childhood, for example. But I think the expense of conducting decent quantitative research to establish this link would outweigh the benefits of evidencing something that already seems pretty well-evidenced and accepted by individual accounts from trans people's recollections of their own childhoods.
Also, we would have no control group of children raised in a non-sexist society to compare the subjects to, as children receive messages from multiple sources (friends, school, tv, adverts, etc) in addition to their parents. One can make simple inferences from looking at different countries with different values/lifestyles and seeing if they have the same proportions of trans-identifying children without needing to conduct a study, though.
I think probably the simplest research would be a qualitative study of the available memoirs/interviews of trans adults to look at their explanations as to how they knew they were trans, and see how many rely on sex stereotypes (spoiler alert: most of them).