How does an MCU detect current drop in an active-low button?
When the switch is open (like in the image), there is some current, I, flowing into the MCU.

Yes.

But when the switch is closed, wouldn't the current just flow to both the GND and the MCU like in a parallel circuit?

No. When the switch is closed, there is a short between the mcu pin and ground. So current will flow out of the mcu to ground, and through the resistor to ground.

If so, how does the MCU detect when the button has been pressed if there is constantly current flowing into it?

Current flows into the MCU when the switch is open, and the voltage on the MCU pin is high. Current flows out of the MCU when the switch is closed, and the voltage on the MCU pin is ground.
Most MCU's these days are CMOS, and the current into or out-of a CMOS input (in steady state) is quite tiny. It is the voltage at the gates of the input mosfets which determine whether the MCU is detecting a &quot;high&quot; or a &quot;low&quot; input.