These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
The parents. In both movies, they were people who got divorced and, to make it easier on themselves, decided that each would only get custody of one child and never mention the existence of their sibling.
Hallie in the remake comes quite close as well. She initiates all the arguments (pulling Annie into the water, stealing her clothes), behaves like a spoiled brat in the isolation cabin and forces Annie to deal with all the drama surrounding Nick's relationship with Meredith because she wants to spend more time with her mother. And there's the fact that Annie is forced to both cut her hair and get her ears pierced, all Hallie has to do is change her accent.
Ear Worm: The original movie has "The Parent Trap" and "Let's Get Together," two of the first songs that The Sherman Brothers wrote for a Disney movie.
Family-Unfriendly Aesop: Children of divorced parents can, through hilarious hijinx, remind their parents that they love each other and get them back together. Therefore, you children of divorced parents out there in the audience, if your parents remain divorced, it's your fault for not trying hard enough to get them back together.
Fridge Horror: Exactly how many relatives were aware of what the parents had done and were forced to keep it a secret? Even the servants and staff knew and had to keep quiet about it.
The Hayley Mills version had three sequels, the last one coming just nine years before the Lindsay Lohan version. Anyone remember them? The third and fourth got hit especially hard; The Parent Trap II at least became bundled with the original on DVD.
On the other hand, whether you're more familiar with the Hayley Mills version or the Lindsay Lohan version will probably depend a lot on how old you are.
Tastes Like Diabetes: Averted completely with the classic original. While the remake feels very much like a Hallmark movie, and Hallmark has played it a lot, it has an undeniable charm that prevents it from being your stereotypical Hallmark flick.
The idea that parents who broke up in a divorce should get back together, no matter how much they fight is odd to 21st century society.
Mostly avoided in the remake. It's mentioned that the main reason for the divorce was "we were both young, we both had stupid tempers" as well as Elizabeth technically not wanting to leave — she expected Nick to stop her so they could make up but Nick believed she wanted to leave. Now that they've matured and sorted their issues out, they might fare better.