If you have 8 parallel strings of 3 LEDs (i.e. each string is composed of 3 LEDs wired in series), then the total current is being shared between the 8 strings. For convenience, we'll assume that the current is shared equally, but this is some difference from string to string.
Each string gets 1/8 of the 3.9A, or 0.49A. Each LED is seeing 1/3 of the voltage (approximately) that is measured across the paralleled strings, which is 9.65/3, or 3.2V.
Did we discuss the fact that the voltage and current of the LED have a unique relationship that is usually defined in a graph? Actually, all diodes have similar relationship between voltage and current, although the scaling will vary.
Here's the graph from the wiki page for diodes....
https://en.wikipedia.org/wiki/Diode#mediaviewer/File:Diode_current_wiki.png
In the part labeled "forward", the diode is conducting. When the current is small, the voltage increases rather quickly. ...or to put it another way, if you are controlling the voltage across the diode, the current increases very slowly, but then increases a lot once you hit 0.6V or so. With white LEDs, that voltage is around 3V.
Once the diode is conducting in the forward direction, the voltage does change as the current changes, but not a lot.
In your case, as you change the wire resistance, you change the current going to the LEDs, which then causes the voltage across the LEDs to change a small amount, which also changes the current.
There must be a video somewhere on YouTube where Ohm's law is described, where circuit analysis and Kirchoff's current law is described, etc. You really should have some understanding of electronics at a fundamental level to be working with this stuff.