We　theoretically　study　the　interplays　of　electron-phonon　interaction,　inter-dot　coupling　tunneling　and　intralead　electron　interaction　on　transport　properties　of　parallel-coupled　double　quantum　dots　connected　to　Luttinger　liquid　leads.　The　nonequilibrium　Green＇s　function　method　is　used.　Numerical　results　show　several　intriguing　physic　phenomena.　In　the　weak　intralead　electron　interaction,　there　is　a　zero-bias　antiresonance　dip　followed　by　phonon-assisted　satellite　peaks　for　strong　electron-phonon　interaction　and　inter-dot　tunneling.　With　decreasing　inter-dot　tunneling,　there　is　an　evolution　from　satellite　peaks　to　dips,　which　signifies　the　phase　transition　from　phonon-assisted　single-channel　to　two-channel　Kondo　physics.　When　the　dot-lead　coupling　is　asymmetric,　in　the　case　of　moderately　strong　intralead　interaction,　strong　electron-phonon　coupling　and　weak　inter-dot　tunneling,　there　may　appear　the　pronounced　negative　differential　conductance　and　Fano　resonance,　while　there　does　not　when　the　coupling　is　symmetric.　These　results　provide　a　convenient　way　to　probe　destructive　quantum　interference　and　extract　information　of　the　electron-phonon　interaction　strength,　which　is　crucial　for　designing　molecular　switches　and　thermoelectric　devices.