Flow in parallel paths after pressure regulator Consider the following:
                               /‾‾ Valve --- Watering hose
Pump --- Pressure regulator ------ Valve --- Watering hose
                               \__ Valve --- Watering hose

A pump generating an arbitrary pressure and a pressure regulator
to curb it at 1.5 bar. After the regulator we have three valves in parallel.
If I close one of the valves partially or completely will the volume
flowing through the other two valves change? I hope that the pressure
regulator would compensate closing/opening valves by keeping the pressure
constant but I'm not sure. 
Background is a watering solution in gardening. I aim to decouple several
strands of watering hose so they can be adjusted independently.
 A: 
I hope that the pressure regulator would compensate closing/opening valves by keeping the pressure constant but I'm not sure. 

In the absence of confounding factors, this is the correct viewpoint.
In order for the pressure regulator to be "perfect", you would ideally have a large volume between the pressure regulator and the valves, and the valves would be spaced a good distance away from each other.  In the current picture, the pressure regulator controls the pressure by controlling the flow from the pump.  At least this is my understanding.
Non-ideal conditions will be seen when the interacting kinetic forces between the different valves are significant.  In that case, the pressure that pressure-regulator sees can't exactly be called the back pressure behind each valve.  If we treat the region up to the valve as a stagnant, constant pressure, region then there is no problem with your view.  We would also require that the pressure-regulator performs its job perfectly, obviously.
note: I'm assuming the pressure-regulator controls the pressure with its own valve.  The representation could be trying to imply that the pressure-regulator is just a sensor and the controller opens and closes one or more of the hose valves, but I don't think this is the system because the OP implies that the hose valves are manually controlled
