I foolishly pulled 60' of 10/2 cable only to open the cover of my panel to find it has no room for the needed 30 amp breaker. Can I pigtail the two 14 ga wires from two 15 amp breakers with one of the 10 ga wire conductors?
Don't do that! What you're proposing, using two 15A breakers as a substitute for one 30A breaker, is called Circuit Breakers in Parallel and is the topic of
240.8 Fuses or Circuit Breakers in Parallel. Fuses and circuit breakers shall be permitted to be connected in parallel where they are factory assembled in parallel and listed as a unit. Individual fuses, circuit breakers, or combinations thereof shall not otherwise be connected in parallel.
Paralleling is only allowed for really big connections - typically 100s of Amps, where otherwise wires would be so large that installation would be difficult and the wires very expensive.
What you may be able to do is replace the pair of 15A breakers with a double-pole 30A breaker. Why double-pole? Most 30A devices are actually 240V, not 120V. If that's the case, you actually do need breakers, but they need to be 30A, not 15A, and they need to be common-trip, which means a double-breaker rather than 2 separate breakers.
10/2 is black & white. While generally speaking black goes on hot and white goes on neutral, pure 240V circuits are an exception. Black goes on one breaker and white goes on the other breaker. No neutral because neutral (in a US 240/120 system) is only used for 120V circuits.
It sounds like you may be trying to do something extra dodgy! Don't!
You can't do that. But that doesn't make any sense. If you have two 15A breakers unused for anything, why wouldn't you just pull them out and replace with a 30A breaker?
If the circuits are still being used for other things, then again, stop and think. If the other loads on the circuits are so small that the 30A thing can use most of their capacity, then those two 15A breakers could simply be combined onto one. Pigtail the two #14 wires to another #14 or #12 wire and land that on one 15A breaker. Now you have an unused breaker; swap that for your 30.
Other than that, I have to say, you are asking an XY problem. That is, you have a problem, but you're not asking about how to solve the problem, you're focusing on a method - and not surprisingly since you're a novice, it's not well-chosen. I recommend you re-ask the core question of "I need to add a 30A circuit breaker to may panel for ______ load (do say), but my panel is full, what do I do?" We answer those all the time.