GUI to delete parts and only person can see it, help?

Is your game filtering enabled (in other words, experimental mode off)? When filtering is enabled, any actions performed in the client do not replicate to the server and are not shown to other clients (with a few exceptions). Turning this on would make your code only delete the part in the workspace named Extra on the client. To turn it on, click on the Workspace object and set the "FilteringEnabled" property to true. After turning this on, your code would only destroy it for the player who pressed the button.

If you want it to be destroyed for all players, you would have to use the following code to do such that.

-- Script in ServerScriptService
local remote = workspace.DestroyPart - Create a RemoteEvent named DestroyPart

remote.OnServerEvent:Connect(function(plr)
    game.Workspace.Extra:Destroy()
end)

-- Client Script
local remote = workspace.DestroyPart
script.Parent.MouseButton1Click:Connect(function()
    remote:FireServer()
end)