Why do bulbs glow brighter when connected in parallel?

The bulbs will only appear brighter if the available current to the system is not limited. In that case the series bulbs will have a lower voltage across each individual bulb and they will appear dimmer. If the power input to the circuit is a constant than the total wattage output from all bulbs is also constant and the bulbs will all appear the same (assuming the filaments for the bulbs are all identical resistance).

In a typical simple circuit the power source will be a battery which attempts to hold a constant voltage across the circuit. In this case the voltage across the bulbs in parallel will be equal to the voltage of the battery and the current through the bulb will be defined by $V = IR$ where $R$ is the resistance of the filament. This means more current (and thus more power) will be drawn from a battery into the parallel circuit than a series one and the parallel circuit will appear brighter (but will drain your battery faster).

I crafted this answer for this question in the first place but since it got closed, I will post it here to at least contribute.

1) The brightness of a light bulb depends on various parameters, most of them being intrinsic properties of light bulbs. Essentially, the brightness depends upon the luminous flux of the light source. However, light sources which emit light with different wavelengths but same luminous flux can be perceived to have different brightness levels. Therefore, luminous flux is useful if we are comparing the brightness of light sources which emit light with same wavelength.

For incandescent light bulbs, brightness or luminous flux is directly related to the heat energy due to the flowing current in a conductor since these type of light bulbs are used by heating the filament until it emits visible light(assuming we have an incandescent light bulb here because other light sources like LED will have different properties). What is the term used to specify the heat energy generated by the flowing current per unit time? Power. Therefore, we should increase the power due to a current source as much as possible to increase the brightness of the light bulb. To find which parameters we should play with to increase the power, we can use Joule-Lenz law which states that: $$ Q\propto I^2Rt $$ Therefore, since power is $\frac Wt$, we can derive the expression that is proportional to the power: $$ P\propto I^2R $$ However, this expression can deceive you to think that increasing the resistance of the light bulb increases the brightness. Since altering the resistance will also decrease the current passing through the light bulb and even exponentially decrease the power, we can derive a more reliable formula by using the specialized form of Ohm's law($V=IR$). Assuming we have an ideal conductor here, one can find that; $$ P\propto VI $$ Overall, you need to increase the emf of the current source to increase the brightness of the light bulb.

2) As all answerers pointed it out, when we wire light bulbs in parallel instead of in series, we decrease the equivalent resistance of the circuit; and therefore increase the current passing through the filaments of the light bulbs. This leads to more power each light bulb is getting(due to Joule-Lenz law) and brighter light bulbs.

Think about the power supplied to the bulb. Assume for a moment constant voltage source, and constant resistance for each bulb (not true for bulb but often used to simplify discussion at this level) then in series you have a total resistance of $2R$ and power $P= VI = \frac{V^2}{2R}$ . This power is split by two bulbs so each sees $V^2/4R$. When the bulbs are in parallel, each bulb sees the full voltage $V$ so $P=\frac{V^2}{R}$. Since a bulb glows brighter when it gets more power the ones in parallel will glow brighter.