Please note that I'm writing from my six year old perspective. So if I say something is hard, it's not hard for me anymore. So don't think I still struggle with any of these hard parts, lol.
Mt. Moon and Rock Tunnel
For some reason, I could never find my way through either caves. My heart would start racing whenever I saw the exit, too.
Brock on Yellow
I was six at the time and I haven't played Yellow since, and I didn't know that Mankey or Nidoran had a type advantage over rock. So I spent almost all of my time grinding south of Pewter until Pikachu knew Quick Attack. Once then, I'd pray for my battle against Brock. I don't remember ever getting past Lt. Surge, though.
GARY MOTHER****ING OAK
Although now it doesn't bother me, as I know when he appears at all times, but when I was younger, I wasn't entirely sure when or where. And usually my favourite thing to do was explore the area before doing what I was supposed to be doing. So naturally, Gary would come up when I least expected him, destroying my weakened team.
The Silph Co.
I've always hated the Team Rocket parts the most. It just drags on. Walk for a little bit, then fight a Rocket Grunt. And the puzzles were mindnumbingly difficult when I was six. Now they're easy, but repetitive. :/
Sabrina and Lt. Surge's Gym
I'm sure a lot of other people hate these gyms too. Pretty much any gym with a random puzzle seems to get on my nerves, but these two take the cake. It took me at least an hour to figure out how to open Lt. Surge's door, and even longer to get to Sabrina and fight her, then to get out.
Surfing anywhere
I've always detested surfing. It's not that it's hard to do, but... Wild TENTACOOL appeared! Over and over and over again.
The relentlessly long routes that are packed with trainers
Usually it seems really long when my team is weakened, though.
One hit KOs
Especially when you're at full life your opponent gets a critical hit.
Trainers who use recovery items
I know it's a valid strategy, but it still sucks, bringing a Pokémon down to it's knee. The feeling of having victory taken away for a few turns is annoying. "You're just postponing the inevitable, Mr. Oak!"