"advertising", "partnering" & "sponsorships" all that really means IS YOU pay them an amount of money TO use there logo's/products or whatever.
I know because both my work "partners" and "sponsors" football clubs and such.
I could post you a nice email form my work address with all the lovely logos of our partners (which we "paid" for).
Like you said Disney is a media company (ok its a boring media company but it still is) would they let anyone use there "brand name" just for fun without a "small" fee after all its American and a Big business since when did they give anything away for free?