It might be considered parody, if it's actually parody, otherwise it's an infringement of the songwriter's exclusive copyright in the music and lyrics. Parody doesn't mean using someone else's work because you want to use an already well-known song to adapt to some new use -- a legal parody is one where the new lyrics actually parody the original work.
The facts that 1) you're not charging admission may affect the owner's damages, and that 2) your according "inspired by" credit, don't shield you from infringement if it's infringement.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
I start from the premise that, under the circumstances you describe, it would be lawful for you to perform the song just as it was written without any license or permission from, or any attribution to, the owner of the copyright in the song.
Copyright law expressly permits your non-licensed use of the song. See 17 USC 110 ("Notwithstanding the provisions of section 106, the following are not infringements of copyright: ... (4) performance of a nondramatic literary or musical work otherwise than in a transmission to the public, without any purpose of direct or indirect commercial advantage and without payment of any fee or other compensation for the performance to any of its performers, promoters, or organizers, if ... there is no direct or indirect admission charge ... .").
Because you can lawfully perform the song as written without a license, I think you have free rein to change its lyrics and perform the song as modified. There is no need to engage in any "fair use" analysis (which includes the inquiry that you're focused on -- that is, is your modified song a parody).