How to make a script only affect the local player parent?

Well, I wouldn't recommend putting that MouseClick event inside the PlayerAdded event... You can easily grab the player who clicked in the MouseClick event because MouseClick returns Player! Read about it here

With that being said, here's your fixed script;

local MS = game:GetService("MarketplaceService")
local VIPID = 8110615
local clickDetector = script.Parent.ClickDetector
local RS = game:GetService("ReplicatedStorage")

game.Players.PlayerAdded:Connect(function(plr)
    if MS:UserOwnsGamePassAsync(plr.UserId,VIPID) then
        --not sure what you want here
    end
end)

clickDetector.MouseClick:Connect(function(plr)
    plr.leaderstats.Potatoes.Value = plr.leaderstats.Potatoes.Value + 1
end)

I would say handle the click on the client, then fire a RemoteEvent. Change the player's leaderstats on the server then.

Use a Local Script

@GeneratedScript

I tried to do that and this is my script, but when I try to write plr it underlines it red?

game.Players.PlayerAdded:Connect(function(plr)

script.Parent.OnServerEvent:Connect(function()
    plr -- Here
end)
end)