Neon lamps in parallel DC
Neons need more voltage to strike, than to maintain, as you have measured. The first neon to strike will drag down the common terminal to less than the striking voltage for the second.

Ideally reduce the value of the series 10k, and increase the value of the 4.7k resistors. 

I wonder why you are using the two 10k resistors at all? Are you trying to switch these in shunt, that is drag the common node down with a transistor to turn the neons off? If so, that is the wrong topology to use, as reducing the series 10k to get both neons on will increase your standing current.

Most people use an NPN switch between ground and one neon pin, with the other tied to a stiff rail through a current limiting resistor. This topology has the added advantage that the switching transistor can be lower voltage than the full rail, as the neon isolates some of the drop.