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We present a linearized phase-modulated analog photonic link that utilizes parallel interferometric demodulation to suppress the third-order intermodulation distortion present in the link. By utilizing two parallel interferometers with different differential delays, we are able to balance the third-order distortion in the output photocurrent and achieve a fifth-order limited link response. We demonstrate shot noise-limited spurious-free dynamic ranges of ~ 130 dB (1-Hz bandwidth) at ~ 1 GHz for a total DC photocurrent Idc~ 8 mA and ~ 131 dB (1-Hz bandwidth) at ~ 5 GHz for a total DC photocurrent Idc ~ 7 mA . These results represent a nearly two order-of-magnitude improvement in dynamic range over conventional intensity- or phase-modulated analog photonic links for a fixed received photocurrent.