For deck stairs adjacent and parallel to the house wall, do I need handrails on both sides?
No, one handrail is required on stairs with 4 or more RISERS. (R311.7.7.) it needs to have a minimum of 1 1/2" clear from the handrail to the wall and shall be mounted not less than 30" nor more than 38" above the nosing of the tread. (R3126.96.36.199.) The handrail shall be continuous, but it can be interrupted by newel posts at turns. (R3188.8.131.52.)
The size is complicated, but don't go larger than 2"... (R3184.108.40.206.)
AND, all stairs shall be lighted. (R311.7.8.)
Below is the Ontario Building Code section. I will mark as valid Lee's answer since he was right and he could not know where I live. The probability is that more US folks will read this so that is the generally accepted answer for both US and Canada
220.127.116.11. Handrails (1) A stairway shall have a handrail on at least one side, and if 1 100 mm or more in width, shall have handrails on both sides.