We present the annual trigonometric parallax of PSR J1744-1134 derived from an analysis of pulse times of arrival. The measured parallax, pi = 2.8+/-0.3 mas ranks among the most precisely determined distances to any pulsar. The parallax distance of 357+/-39 pc is over twice that derived from the dispersion measure using the Taylor & Cordes model for the Galactic electron distribution. The mean electron density in the path to the pulsar, n_e = (0.0088 +/- 0.0009) cm^{-3}, is the lowest for any disk pulsar. We have compared the n_e for PSR J1744-1134 with those for another 11 nearby pulsars with independent distance estimates. We conclude that there is a striking asymmetry in the distribution of electrons in the local interstellar medium. The electron column densities for pulsars in the third Galactic quadrant are found to be systematically higher than for those in the first. The former correlate with the position of the well known local HI cavity in quadrant three. The excess electrons within the cavity may be in the form of HII clouds marking a region of interaction between the local hot bubble and a nearby superbubble. Comment: revised version accepted for publication in ApJ Letters; reanalysis of uncertainty in parallax measure and changes to fig 1