You do need connections in the industry, recording, publishing, management, etc. You can parody anything you wish for exposure but you can't sell it without making an agreement with the original publisher of the song.
I've written many parodys but I don't put them on any market for sale. Writing original comedy songs is much easier to market. You don't have to worry about cutting the royalties. If you can write parodys, you can write originals.
I would suggest getting involved with a local songwriting workshop. NSAI (Nashville Songwriters Association International) has many such workshops around the country. They can give you guidance and important information.
As you hone your craft, learn everything you can about the business. The best of luck to you.