Also, unrelated, but you should be setting .Parent last.
So ideally, your server code should be this:
--Function that fires the event
local Players = game:GetService("Players")
Players .PlayerAdded:Connect(function(player)
player.CharacterAdded:Connect(function(character)
local root = character:WaitForChild("HumanoidRootPart")
local prompt = Instance.new("ProximityPrompt"):Clone()
prompt.Name = "PairPrompt"
prompt.ObjectText = player.DisplayName
prompt.ActionText = "Pair"
prompt.RequiresLineOfSight = false
prompt.Parent = root
end)
end)