However the 'expert' she consulted, one Roderick P. Hart, who is a professor of communication at UT Austin, made a fairly boneheaded comment:
"It's a noun used to modify a noun, and everyone knows you use an adjective to modify a noun."
Given the way traditional grammar is usually mis-taught, that idea is understandable. In that scheme, almost anything that modifies a noun is called an adjective. But I'm afraid it just isn't so. Nouns frequently modify other nouns, and our expert has just fallen into the common trap of trying to use (mistaken) grammar to give an it-must-be-so explanation.
a leather wallet (cf., a wallet of leather)
the White House spokesman (cf., the spokesman from the White House)
The Cambridge Grammar doesn't seem to go into this issue in much detail, but the older A Comprehensive Grammar of the English Language does (see 17.104-17.109).
'Democrat Party' doesn't seem all that unusual in grammatical terms. Most noun + noun phrases have paraphrases with prepositions, as in the examples above.
So the 'Democrat Party' comes from "the party of Democrats."
True, the s vanishes in the attributive noun, but that is normal. The number of the noun becomes neutralized in most cases. Hence (examples from the Comprehensive Grammar):
the picking of hops ~ hop picking
decay of teeth ~ tooth decay
Although there may be some constraint I'm unaware of, the thing that really makes 'Democrat Party' seem odd is that there is a well-established adjective form, Democratic, and when you make a distinction like that, people are inclined to assume that you are doing so to highlight a semantic difference. In this case that the Democratic Party is not all that democratic.
Which is the main thrust of the article.
I also note that the equivalent option to try to get in a sly dig at Republicans is not available because there's no morphological difference between adjective and noun:
the party of Republicans ~ the Republican party
Deleting the final suffix yields "the Republic party", which I don't think would be interpreted as applying to Republicans.