How to make falling snow from hundreds to thousands of particles without lag?

Is it also possible to make this with GUI or local parts?

A good solution would be to generate the snow particles locally, around the local player. This way you don't generate particles the players won't even see.

If you're using particles, it might be as easy as making a part the releases particles follow the player on the Y axis, above their head.

-- local script in StarterCharacterScripts
local snowParticlePart; -- define this, or create one via the script

local player = game.Players.LocalPlayer
local character = player.Character or player.CharacterAdded:Wait()
local head = character:WaitForChild("Head")

local heightAboveHead = 20
while true do
	snowParticlePart.CFrame = CFrame.new(0,heightAboveHead,0)+head.Position
	wait(.5) -- probably doesn't need to be any faster
end

Just make sure LockedToPart, a property of ParticleEmitters, is false.

This solution works best with FilteringEnabled.