The answer is that spondylolisthesis of L5-S1 will not cause neck pain by itself but may magnify preexisting problems in the neck. If you need to alter your gait to compensate for the spondylolisthesis and have a degenerative neck, this may magnify neck problems.
I worry about your “weakness” from the spondylolisthesis. This condition can produce bone spurs that can compress the nerves adjacent to the pars fractures (see isthmic spondylolisthesis on website). If there is weakness present from nerve compression and not pain inhibition- surgery needs to be considered as nerve do not recover easily from compression.
Dr. Corenman