A current in both loops
Adding a second bulb in parallel to the first sets up a second circuit loop in which charged particles can be set in motion. Not only do we have the loop of charged particles passing through the first bulb, but we also have a second loop of charged particles flowing through the second bulb. The number of charged particles passing through the battery each second is thus doubled, with equal contributions from each separate, independent loop.
In other words, the current in the battery is double that with one battery and one bulb, whilst the current in each bulb is the same as in a circuit with one battery and one bulb.
In each loop of the circuit, the charged particles are set into motion by the battery and are shifting energy to the surroundings as they pass through the filament.
Two bulbs of normal brightness
When a second bulb is added in parallel the electric current in each bulb is the same as the circuit with one battery and one bulb. As the loops are independent, each functions exactly as the single loop. The two bulbs are symmetrical, and so both bulbs are of equal brightness (and of the same brightness as one bulb in a single loop, precisely because that is what they are).
Connecting bulbs in parallel does not get you something for nothing, even though at first sight this might appear to be the case! The good news is that we have two bulbs of normal brightness. The bad news is that the energy in the chemical store of the battery is depleted twice as quickly as with one bulb. The electric circuit model confirms this point of view because the electric current in the battery (the flow of charged particles shifting energy) is doubled as the second bulb is added in parallel.