Horizontal tangents are such that dy/dx = 0; this happens when
(y - 3) / (x - 3) = 0  →  y - 3 = 0  →  y = 3
so (I) is true.
Vertical tangents are such that their slope is undefined; this happens when the denominator vanishes, or
x - 3 = 0  →  x = 3
so you are right that (II) is not true.
To assess whether (III) is correct, consider what happens at different nearby points (with neither x = 3 nor y = 3) where the y coordinate is kept the same so that the tangent lines occur in the same row. For example,
x = 0, y = 0  →  dy/dx = (-3)/(-3) = 1
x = 1, y = 0  →  dy/dx = (1 - 3)/(-3) = 2/3
x = 2, y = 0  →  dy/dx = (2 - 3)/(-3) = 1/3
The slopes are not the same - they have to be if these tangents are supposed to be parallel - so (III) is not true.
This makes A. (I) only the correct choice.
