I don't think any extremist murderer represents any specific party either.
But many posters here absolutely believe that any extremist murderer with any marginal right-wing views represents the Republican party in its entirety... and that any extremist murderer with any left-wing views no matter how blatant is just a single crazy person and it has nothing at all to do with politics. Many posters here express the view that Republicans are entirely 100% at fault and to blame for any violence committed by anyone with any conservative leanings - any right wing violence is directly caused by Republicans, and supported by and incited by Republicans. On the other hand... Democrats are completely blameless and have absolutely no role in any left wing violence that is committed, regardless of any related factors. Democratic politicians spend a decade telling everyone that Republicans are fascists, and the people need to take action to stop them, Republicans need to made unwelcome in society, and implying that conservative citizens across the board are nazi supporters... and when someone references fascism in their politically-motivated assassination, somehow Democrats had nothing at all to do with that, and have no responsibility for the rhetoric they've been preaching.
By all means, hold Republicans responsible for the abhorrent things they say, and the way they've contributed to schism and division in the US. Absolutely.
But don't excuse Democrats from responsibility for the ways they've incited political strife and riled up citizens.