I know what a normal suspension is and how it resolves downwards.
Is a double suspension when a suspension occurs in two of the parts?
But what is a triple suspension? Is that when it resolves upwards instead of downwards??

Answer to the first question: Yes.

Answer to the second question: The only example of triple suspension that I can think of is at a perfect cadence. Here is an example:

Soprano = D down to C
Alto = B up to C
Tenor = F down to E
Bass = staying on C

The above notes would be heard after a dominant seventh chord. The triple suspensions result from the carry-over of the notes in the dominant seventh chord, and they delay the notes of the tonic chord. Note that the leading note B in the alto resolves upward to the C, while the soprano and tenor resolve downward, just like when you are writing a simple V7-I progression.