I plan to self-publish for profit a book that playfully parodies famous Xmas songs. Is copyright infringement an issue?

Parody is a form of fair use, which is a defense to a charge of copyright infringement. The test you have to meet is very fact specific, and requires your parodies to "transform" the original works enough. How much is enough? It just depends. Are you commenting on the original songs, or just using well known melodies and most of the words to get your dog-themed works some attention? If so, pay special attention to the Dr. Suess case below.

You might want to review some recent case law, all of which are much discussed online, including:

Salinger v. Coltin, the "Catcher in the Rye" sequel about Holden Caulfield, 60 years later, enjoined by a New York District Court judge as not being a parody. I haven't read the new book (I can't, since it was enjoined), this sure sounds like it should meet the test for parody.

Dr. Seuss Enterprises v. Penguin Books USA, case (11th Circuit, finding infringement in the use of Dr. Suess -style rhymes to tell the O.J. Simpson story).

"The Wind Done Gone" case, SunTrust Bank v. Houghton Mifflin Co. (11th Cir., finding non-infringing parody of "Gone with the Wind")

Thje "Pretty Woman" case, Campbell v. Acuff (Supreme Court reversing the 6th Cir.'s finding of no parody and remanding for further proceedings, saying basically that maybe there was parody in 2 Live Crew's song mocking Roy Orbison's song)

The "Barbie Girl" song, ruled a parody by the 9th Cir. Mattel v. MCA.

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.

Copyright infringement is definitely an issue that you need to think about, but there are two reasons to think that much of what you are doing will not be infringing.

Public domain. I do not know off hand the date of publication for most of the original songs that you are parodying. But, if they were first published before 1923 in the United States, then they are now in the public domain, meaning there is no more copyright protection for those songs. Because the term and requirements for copyright have changed over the years, there may be newer songs that are in the public domain, but generally if its before 1923 it is public domain.

Parody. The rap group 2 Live Crew did a parody of Roy Orbison's "Pretty Woman" that the Supreme Court ruled was a fair use and therefore not an infringement. I can't tell just by the titles of the songs if they are fair use parodies but the very well could be. The problem with relying on fair use is that you can rarely be entirely sure if you are in the clear. Each case is different. The Copyright Office has a pretty good set of fair use guidelines.

It would probably be useful to get an opinion from an attorney who sees the full work. It could probably be done in an hour or two. Just keep in mind that any attorney's opinion on these issues is not likely to a guarantee that you are not infringing.