Help with parametric formulas?

So I was given x = t^3 - 3t and x = t^2 - 8.

I needed to find the points along the parametric curve where there were horizontal and vertical tangents.

I. Horizontal:

(Step 1: Set dx/dt = 0 to find t)

dx/dt = 3t^2 - 3

0 = 3t^2 - 3

3 = 3t^2

1 = t^2

t = +/- 1

(Step 2: Check that dy/dt doesn't equal 0 for the t-values, and find the x- and y-values for each t-value by plugging t into the original equations for x and y)

dy/dt = 2t doesn't = 0 @ +/- 1

t=1:

x = 1-3 = -2

y = 1-8 = -7

t=-1:

x = -1+3 = 2

y = 1-8 = -7

(Step 3: State your points where the curve has a horizontal tangent)

Therefore, the curve has horizontal tangents at (-2, -7) and (2, -7)

I then applied similar steps to horizontal tangents.

II. Vertical:

dy/dt = 2t

0 = 2t

t = 0

dx/dt = 3t^2 - 3 doesn't = 0 @ 0

x = 0

y = 0-8 = -8

Therefore, the curve has a horizontal tangent at (0, -8).

I thought what I was doing was correct, but when I went to check, I was incorrect in both cases. I feel like I'm missing an important step, but don't know what this could be.