Can Position Sensitive Detectors electrodes be offset? How?
As many suggested - first check that the source is not an optical input from celling lights, but you would see the noise without the resistor as well (assuming conditions were the same).

Your circuit uses NTV0505MC DC/DC converter, it is not an LDO so it produces ripple. In original circuit design the PSRR characteristic of the LTC2055HV suppresses this ripple by a factor of 130dB. When you add the 10M resistor - you inject this ripple straight into the first stage of signal processing line. 10M with 100k of TIA stage forms an "inverting op-amp circuit". So the ripple (or whatever you have on -5V rail) is attenuated by 1/100 factor and added to whatever useful photo-current you get from PSD pin multiplied by 100k TIA feedback. If it is not a DC/DC converter ripple - checkout the shielding of your supply rail. Supply rail may function as antenna picking up EMI emitted by surrounding power wiring. 

Also to me it seems wasteful to use second OpAmp as simple unity gain buffer before the multiplexer. When working with such noisy devices as photodiodes - you would always want to do as much filtering as possible. PSD has large area (so high capacitance), that means with typical TIA design you will not get anything beyond 10kHz from it. So using second OpAmp as 2-nd order 10kHz LPF would be very beneficial to the resulting SNR. 

On my practice I was picking up the PWM noise from brushless motor of the stabilizer gimbal on which the photodiode sensor was mounted. While the whole signal conditioning circuit was shieled inside a metal case, the photodiode casing itself (biased cathode) was exposed outside (or actually inside the surrounding plastic mount of the lens in front of it). With 10MOhm TIA feedback the noise was 10x stronger than the signal itself. Problem was solved with proper shielding to ground of the plastic lens mount using nickel conductive spray paint from MG Chemicals. So the only opening in EMI shield of the whole sensor was an optical lens in front of photodiode.