A musical key can be centered around any pitch and be either major (with a particular pattern of whole steps and half steps between each note in the scale, stereotypically associated with happy-sounding music) or minor (with the third, sixth, and seventh notes in the scale lowered a half step, stereotypically associated with sad-sounding music). For any minor key, the parallel major simply refers to the major key built on that same pitch. The parallel major key of A minor is A major, the parallel major key of D minor is D major, and so on.
You'll generally see a piece written in a minor key shift into the parallel major briefly, for variation, or modulate there towards the end, to signify a shift in mood from sad to hopeful. In the teaching of sight-singing, students' ability to sing in the parallel minor of a given key is an important milestone that marks their understanding of the pitch relationships in major and minor scales.
In education, the phrase parallel majors refers to college majors that share many courses. If two majors offered at the same university are referred to as parallel, this is probably an indication that the university considers them a doable double major. If the term parallel majors comes up in a conversation about a transfer between schools, it's referring to equivalency between a major offered at the old school and one offered at the new school. Even if these two majors have different names, they involve the same topics of study and the same or very similar coursework. If you're changing schools, your credits are likely to transfer and count toward a degree in the new school's parallel major to your major at the old school.
Parallel majors that make good double majors are often two closely related fields for which students are likely to need some knowledge of both in their future careers -- for example, biology and environmental science. Otherwise, usually a parallel major taken on as a double major provides focus and specificity to a student's course of study. For example, the Wisconsin School of Business lists Agricultural Business Management, Strategic Communications and Computer Sciences among possible parallel majors for the business student. A student who double majors in general business and any of these fields would have a specific focus and preparation for a different type of career than one who majored in business alone.