When it comes to those who make it a point to bag on conservatives, I think most people fit into one of these categories:
The Cool Kids
These are mostly true believers who think if you're conservative, you're out of date with current culture and you should be ripped for it. These people are prone to associate conservatives with racists, misogynists, conspiracy theorists and believe conservatives are backwards and unintelligent. They feel good about themselves because they think the same way the really, really cool kids think (i.e., Hollywood, media personalities, popular politicians, etc.).
The Near Cool Kids
These ambitious folks probably don't care one way or the other about most of this stuff, but making fun of conservatives gets them the approval of the Cool Kids. That's really the most important thing to them because it may increase their opportunities. Many in this group also probably enjoy counting their likes and retweets on twitter. Approval, above all else.
The Closeted
These are conservative minded people that don't want to be branded by the Cool Kids as conservative, and therefore be associated with the racists, misogynists, idiots, and conspiracy theorists that the Cool Kids attribute to all conservatives. They have learned that if they make fun of conservatives they usually won't catch as much shrapnel and can generally get along without much risk of being called out, or worse, cancelled.
The Converted
These are conservative minded people who probably remain pretty conservative, but the associations reinforced by the Cool Kids and the Near Cool Kids have started to make them question whether they themselves want to actually be conservative.
And then there is another, more rare group. The Black Sheep.
The Black Sheep cares most about the reaction. Whether that's making someone think (the intellectual), making someone laugh (the comedian), getting under someone's skin (the pest), etc. The Black Sheep is one most likely to make fun of both conservatives and liberals if it suits their purpose.
Now keep in mind that there far more people out there who are conservative or liberal who don't make it a point to make fun of anyone.