Yes! Tastes can definitely change. I love all sorts of things that I used to hate. I often ask picky (adult) eaters if they adored the taste of coffee or beer the very first time they took a sip.
Apparently there are also studies that show that you tend to crave later what you eat now. Or at least not hate it so much. Seems to have a lot to do with your microbiome, which can adjust rapidly.
I listened to a podcast that discussed getting kids used to new foods that they don't initially like by having them eat a pea-sized bite of the food every day for a week or two, with a non-food reward at the end of that time. Just that amount is enough to get the gut bacteria going, "Hey, maybe this stuff is okay after all." Then that information gets passed along to the taste buds, which decide maybe it actually does taste pretty good. I think where a lot of parents screw up (including mine, and I'm sure I would have, too, if I had kids) is by forcing kids to eat, say, a big pile of cooked spinach.
More directly on topic... I hate beets. I've tried them a zillion different ways, and I just can't get past the fact that they taste the way dirt smells. Maybe I need to treat myself like a kid and eat a tiny pea-sized bite every day for two weeks. :-)
I'm not sure what my parents could have done differently, but I was a mess in this department.
As a kid, I was incredibly stubborn about food. I don't know exactly when it started, but as long ago as I can remember (at the very least going back to kindergarten/first grade), 95+% of what I ate was Cheerios, Wheat Chex, skim milk, apple juice, apple sauce, peanut butter sandwiches (no jelly), or pasta with ketchup. The other <5% was occasional things like "snacky" food (candy, individual fruit, etc.), pancakes, french fries, hot dogs, or burgers from McDonald's (and burgers from nowhere else). If a restaurant didn't have pancakes or a hot dog and fries as an option, I basically wouldn't eat there, so I did very very little eating out (that part was a plus, at least, though limiting socially and for family events, which were often just me sitting there with an empty plate or a peanut butter sandwich brought from home).
My parents tried. Sometime around age 8 I at least remember specific times where they'd want me to try "just a bite" of some "new food" every day with my boring dinner. I either refused, choked it down talking about how gross it was, or spit it out (I honestly don't remember the distribution there) for enough days in a row that they gave up. I'm sure there were other attempts along the way, but at some point they basically just let me do my own thing. I prepared (obviously there wasn't much to it beyond putting sandwiches together and boiling water for pasta) all of my own food pretty much from age 10 on as a result. I took a multivitamin every day. Maybe that helped.
This continued for quite a while.
My first sign of progress was at age 19 (yup). I was spending a weekend with some friends in Wisconsin, and we went to Famous Dave's, and I ate a burger from Famous Dave's. That sounds like a pretty pathetic kind of progress, but it was indeed a big step for me. My first ever burger from somewhere other than McDonald's. And I actually enjoyed it. The only reason this happened was social pressure (not deliberate on their part - I had met most of these people online and was meeting them for the first time in-person and most of them as a result had no idea about my food issues).
The about a year and a half later (age 20) I had another breakthrough when I was going to meet my now-GF for steak and eggs at her favorite restaurant near her. I asked my dad about it and he leaped at the opportunity to cook steak and eggs for me. I had it well-done and with ketchup (I've since atoned for this), but I enjoyed it, never having eaten it ever before.
At 21, we bought our first house and lived with GF's sister and GF's sister's boyfriend. Naturally, my life changed a lot. The four of us would often cook together, and while at first it was me standing around while they cooked, I eventually started figuring stuff out. I'd still make peanut butter sandwiches and pasta with ketchup quite a lot for myself, but I ate more and more with them, going from eggs to bacon and eggs to breakfast burritos to baked potatoes to meatloaf to actual fucking vegetables. Later that year, I read some of books and did a lot of research into healthy eating for the first time, making me want to actually be the one to take initiative and try lots of new foods for health reasons.
And sometime at 22, I had my last peanut butter or pasta filled week, switching entirely to eating real food. I became the primary cook in the household. Since that fall, GF and I have been eating what I'll call "vegetable-heavy paleoish" and things have been great. Now there are literally no foods I'm not at least open to trying (I have preferences of course), with the sole exception of anything cheese-based (sorry, still gross if it's present enough that I can taste or smell it - small doses of shredded mixed into something can be ok).
So tl;dr: factors that went into fixing my stubborn/picky eating habits:
I grew the fuck up.
Social pressure from friends unaware of my issues.
A woman.
Learning about healthy eating.
Cooking my own food.
It's possible my parents could have inspired some of it had they done things differently, but I really have no idea what could have worked.