but then I don't really get how a 7-segment LED could work.
Because you have to have a resistor in each of the LEDs.
I have one that has a common anode to which you attach one resistor,
If you do you have the same problem, the segment's brightness is dependent on how many segments are on.
So how can the led's in this display work in parallel with just one resistor?
It can't. Not properly.
According to the datasheet there don't seem to be any internal resistors?
Yes but why should there be any internal resistors in this part? I have never known a 7 segment display with built in resistors. Remember resistors are only one way of limiting the current, you can also use a constant current driver.