The Legend of Zelda entered the 16-bit era on the Super Nintendo in 1992 with the release of A Link to the Past (1991 in Japan), and despite having the most dungeons of any Zelda game, A Link to the Past is also an oddity because of its unique design for Link - who has pink hair. Link is typically depicted as either a child or young adult clothed in green with blond hair. The Link in ALttP still wears the iconic green tunic, but for some reason has bright pink hair.
Each game in the series has a distinct style, but Link has aesthetically almost always remained the same. Occasionally he'll have multiple outfits, or even a wide range of them like in Breath of the Wild, where his primary outfit is actually blue, and inspired by The Winder Waker's lobster pajamas. His hair, however, is almost always some shade of blond. Sometimes it's a very bright yellow like in Wind Waker, or a darker blond like in Twilight Princess, but A Link to the Past is the only game where it's been pink.
The strangest part of the situation is that it appears Nintendo has never given a rationale for Link's pink hair in A Link to the Past. Adding to the confusion is the official Nintendo instruction manual for ALttP, which has artwork depicting Link with his usual blond head. It seems that canonically, Link still has blond hair in A Link to the Past, but it was changed to pink for the in-game sprite for some reason. Unfortunately, there may never be an answer unless Nintendo decides to provide one, but there are a few theories.
Potential Reasons For Link's Pink Hair In A Link To The Past
One potential explanation for Link's pink hair in A Link to the Past comes from an old forum post on Zelda Universe, dating from 2006. User Snarwin claims the pink hair is a result of the Dark World turning Link into a bunny, and thus there wasn't room in Link's limited, 16-bit color palette for his traditional hair color with the bunny pink taking up a slot. As other users pointed out, though, Link's palette already has yellows and browns in it, and there doesn't seem to be any reason two separate sprites would have to share a color palette.
Others posit that it was simply to make Link's hair stand out from the rest of his sprite. Playing A Link to the Past on an old CRT TV in 1992 might have made it hard to get a good look at Link if his face, hair, and hat made a rough gradient from skin tone to green. Some players, such as a couple of different users in a Reddit thread from 2011, theorized that Link's hair was supposed to be brown, but would have blended in too well with the environments, so it was changed.
This is at least plausible, since the earliest versions of Link do appear to have hair very close to a shade of brown. Even his hair in the A Link to the Past manual looks like a light brown when compared to Zelda's bright blonde. Link wouldn't have definitively blond hair until Ocarina of Time in November of 1998. Roughly a month after Ocarina dropped, a color version of Link's Awakening came to the Game Boy Color, and the Link sprite appears to have hair in a shade closer to brown.
According to the instruction manual for A Link to the Past, it appears that Link's hair is canonically light brown, or dark blond. At some point between then and Ocarina of Time, Nintendo decided to make his hair more brightly blond. For some reason - whether technical limitations, or a design decision - Link's sprite in The Legend of Zelda: A Link to the Past was given pink hair. Unless Nintendo decides to divulge why, it may remain a mystery.
Sources: Nintendo, Zelda Universe, Reddit