how does a parallel circuit change when a branch is added?\no the total resistance increases, so the current…

Brief Explanations:

In a parallel - circuit, the formula for total resistance $R_{total}$ of $n$ resistors $R_1, R_2,\cdots, R_n$ is $\frac{1}{R_{total}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots+\frac{1}{R_n}$. When a new branch (a new resistor) is added, the sum on the right - hand side increases, so $\frac{1}{R_{total}}$ increases, and $R_{total}$ decreases. According to Ohm's law $I = \frac{V}{R}$, with voltage $V$ constant, when $R$ decreases, current $I$ increases. Also, power $P=I^2R$ or $P=\frac{V^2}{R}$ (for bulbs), and with $V$ constant and $R$ decreasing, the power of bulbs increases, so they shine brighter.

Answer:

The total resistance decreases, so bulbs shine at a higher brightness.