Why do two different multimeters show completely different resistance readings?
resistor which is soldered into a PCB (forming part of a larger
  circuit)


DMMs measure resistance by applying small current and measuring the resulting voltage. Then assuming that the component is pure passive resistor, the DMM calculates the value in accord with Ohm's Law.

The issue is that when a resistor is a part of a larger circuit, the resulting net contains semiconductor-kind elements and possibly active sources as not fully discharged capacitors. Semiconductor have non-linear dependence on current direction and applied voltage. This would explain one difference. The higher than nominal resistance could be shown if there are active sources that confuse DMM algorithm.

The circuit in question is a 4-channel variable-gain amplifier. It has power supply part that is not shown on schematics:



This hidden part must have bypass capacitors. 

Now, the V2164SZ IC likely has clamping diodes as ESD protection. While the DMM usually applies small probing voltage, 0.2- 0.3V, high impedance measurements require higher voltage. For example, my Fluke 179 applies up to 3.8V in autorange mode. So it is possible that the probing voltage from DMM charges these bypass caps through clamping diodes. Try different directions for different time and different DMMs, and the readings can be anything.

Therefore, don't trust any in-circuit measurement of components without disconnecting at least one end.