The full solution is below, with a short explanation for each group. To avoid spoilers, play the puzzle or open the hints page before reading the answer panels.
PARKA, FROST, ICICLE, and SLED all fit the category "Winter words". Cold weather brings these out. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Coffee shop orders
AMERICANO / MOCHA / LATTE / CORTADO
AMERICANO, MOCHA, LATTE, and CORTADO all fit the category "Coffee shop orders". You can hear these at a counter before 9 a.m. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Hidden numbers
FREIGHT / STONE / TENSION / HEIGHT
FREIGHT, STONE, TENSION, and HEIGHT all fit the category "Hidden numbers". Each word hides a number sound or spelling. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things with strings
VIOLIN / HOODIE / KITE / PUPPET
VIOLIN, HOODIE, KITE, and PUPPET all fit the category "Things with strings". A string is part of the object or how it works. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.