Can I pull a c-wire from a second nearby thermostat?
No, same as any electrical circuit, you need a two way street on the same circuit and the c wire is only one of those lanes. The R wire is the second lane (most cases). Additionally, you risk frying your furnace's circuit board, and your fireplace blower still needs switching that converts back to millivolt.
Your solution is to pull from a transformer (install a low voltage power supply of some kind) and provide an appropriate switch relay.
See here:

Can I share &quot;C&quot; wire on two thermostats?
https://www.google.com/amp/s/www.instructables.com/Nest-Thermostat-With-Gas-Fireplace-or-Other-Milliv/%3famp_page=true