There is no legal reason I know of. I suspect it is to make the names conspicuous. Many contracts were written years or decades ago and have been updated only slightly over the years. Maybe some young lawyer stuck in the back of a dusty law firm capitalized the letters in your form of contract 100 years ago and no one thought to ever change it! Or, maybe the Caps Lock key on his antique typewriter would stick.
Capitalization of a name does not affect your legal rights.
The previous answers have it right. It makes no legal difference and is simply a matter of style. In my contracts and pleadings, I use an acronym for the parties such as the following example: Gulf Coast Manufacturing, LLC ("GCM"). In other words, I define the party first and then use an acronym throughout the rest of the document when referring to that party.