Why does this script keep giving an error of removing a zombie when it dies in my game?

You need to detect if the item is valid for this use FindFirstChild After deleting use break to stop script. You can use this script:

while true do
    wait(0.1)
    if script.Parent:FindFirstChild("Zombie") then
        if script.Parent.Zombie.Health == 0 then
            script.Parent:Destroy()
            script:Destroy()
            break
        end
    end
end

Consider the following code:

local a = (0)

while (true) do
    a = (a + 1)
    wait()

    if (a == 15) then
        script:Destroy()
    end

    print(a)
end

On every wait(), a will increase by a value of 1, until reaching 15 - upon which, the script will be destroyed... however; this occurring does NOT end our loop - because we never broke the loop. So... the actual output of the above code would be (1,2,3,4,5 --> inf). It would never end.

Now, apply the same concept to the code you have provided. Whenever the script detects that the zombie has no health remaining - it destroys the script's parent, effectively destroying the script as well. Our loop will continue to run once this script is destroyed... SO, we are still checking the if statement every 0.1 seconds. You are getting an error because script.Parent is nil now that you have destroyed the script's parent.

To fix this, break your loop when you destroy the script.

while true do
    wait(0.1)
    if script.Parent.Zombie.Health == 0 then
            script.Parent:Destroy()
        break
    end
end