Can I pigtail #10 and #14 wires in my service panel?
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 NEC 240.8:

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 &amp; 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!