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As I read the Chaos codex, it states that warptime can be used in "the start of any player's turn", and it's not a shooting like spell.

So is there a rule to stop a Thousand Sons Aspiring Sorcerer, with his two spells, to cast it on both his and the opposing player's turn?

I am assuming there is a rule against this, otherwise TS troops would be more popular.

Does the codex state that the psychic power can be used at the start of any players turn? Then... you answered your question.

Also, using a power in your turn, and again in your opponent's, does not mean that you've used "two powers per turn" as the Mark of Tzeentch allows. You're allowed two per player turn with the MoT.

The reason TS troops aren't popular (in a competitive sense) is because:

A) No melta access, and only a single psychic power as potential anti-tank (and as someone who used to use Bolt exclusively, I can tell you that it is NOT anti-tank).
B) Paying a premium for a 4++ and AP 3 bolters, but the cover rules make both of these less powerful (as your basic CSM can have a 4+ cover, and enemy SMs will just be downgraded from 3+ to 4+).
C) You must buy an Aspiring Sorcerer who, before choosing your power, costs almost as much as 3 of these overcosted models.

Don't get me wrong, I use TS all the time, because I love the models, love the fluff, love Tzeentch, love the color scheme, and love psychic powers. But there is a good reason they aren't used often.

Yes, warp time can be used in both player turns.

Just for the sake of clarity warptime only affects the caster, not the unit he's in. I get the impression you may be misunderstanding that, apologies if I'm incorrect.