What if I supply 5V instead of 3V?
Then your supply is most assuredly a 20ma current source (if you've previously set current limit to 20mA). If your power supply has a voltage display, it should indicate the LED's forward voltage, since no series resistor is present. A different temperature may change this voltage reading, and a different LED may indicate a different voltage too.
If I wanted to drive both LEDs at 20 mA, at what current would I set my current limiter?
You might try setting current limit to 40 mA. You can be assured that 40mA is emitted by the current-limited supply, but that's dangerous - because one parallel LED might hog most of that current and get hot, while the other LED gets starved.
It is difficult to visually compare LED brightness by eye. Even if two are equally bright, it is not a reliable indication that they share current equally.
Edit:
Some LED data sheets show lumens variation when driven at a specified current:
Note that at 20mA, Luminous intensity can vary 2:1 range from LED-to-LED. In the same batch, you might experience a smaller range than this. Some manufacturers allow binning to achieve a smaller range - but that'll cost you extra.
And also note that forward voltage can vary from 3.0V to 3.6V - and that's at a constant ambient temperature of 25C!
A series-connected string of 70 LEDs ensures current-sharing, but for 3V LEDs, that adds up to about 210 V (awkward). At least 42 W step-up converter would be needed.