If one resistor is added to a circuit bypassing a longer parallel circuit, will the parallel circuit have current?
Not quite. I don't know what the two boxes you made labeled "Power In" and "Power Out" are, but I'll assume they're of no significance.

The circuit you made is equivalent to:







simulate this circuit &ndash; Schematic created using CircuitLab

It should be clear now that you didn't actually bypass the other 6 bulbs. You just added one more parallel path for the current to flow through. Granted, that path will conduct more current than the other two, but it's not a bypass.

I think your drawing is confusing you because it visually looks like the last bulb on the right is bypassing the other 6. But since the red line jumps over the black line at the intersection, it is not a bypass. If you replaced that line jump with a node (and then removed the vertical segment directly below it), it would bypass as you originally stated.