So, just to reiterate. You've got

and

You want to know when you have vertical and horizontal tangents, correct? If so, I'm not sure I would eliminate the parameter. The reason? Because

The derivatives on the RHS there are easy to compute, because you have the formulas right there. In addition, by inspection, you can tell that you'll get horizontal tangents precisely when

and

If they're both equal to zero at a particular value of , you may need to do more work. Also, you'll get vertical tangents when

and

Does this make sense?