Two identical light bulbs are connected to a battery. Will the light bulbs be brighter if they are connected in series or in parallel?

In parallel if the filaments (I'm assuming you're talking about tungsten filament bulbs) will not be burned out by the current in a parallel connection. The reason is that when the bulbs are connected in parallel, the entire voltage is applied to each bulb to drive the current. Whereas, in series, the two bulbs act as a voltage divider in which approximately 1/2 the voltage will be applied across each bulb assuming the bulbs are of the same type. That's, of course, if the bulbs are operating in the approximately linear portion of their characteristic curves.

Not in series as each bulb will only receive 1/2 the voltage!

Parallel is much brighter.

In parallel they each get full voltage and current.

In series they each get 1/2 Voltage and current.

They are brighter in parallel.

in parallel :

brightness (p) ≡ P(p) = V^2/R 

in series :brightness (s) ≡ P(s) = (V/2)^2/R = V^2/(4R) = P(p) /4 

Let's say the resistance of the bulb is R and battery voltage is V.  In series, the resistance of the circuit will be 2R, the current will be V/(2R), and the power at each bulb will be (V/2R)(V/2) = V^2/(4R).

In parallel the resistance of the circuit will be R/2, the current will be 2V/R, and the power at each bulb will be (2V/R)(V) = 2V^2/R.

Brighter in parallel.