Both have their own multiverses.
Most Marvel comics take place in a universe with an Earth called Earth-616. It's the one Earth, out of many. There's an unknown number of parallel realities, and Marvel's never nailed down a limit. Marvel makes use of their multiverse in a few different ways. Some stories are under the Marvel brand, but not on Earth-616, like New Universal. Exiles takes advantage of the multiverse by having people who have been removed from their native realities' timelines, and have to traverse the multiple universes having adventures. New timelines or universes will also be spawned out of various storylines (e.g. The Age of Apocalypse, or Days of Future Past). I've only touched upon this subject, the Wikipedia article goes into great detail of how some universes are parallel timelines, others are alternate timelines that branched off of the Earth-616 timeline at some point in the past or future.
The Crisis on Infinite Earths story brought to an end DC Comics' long-running open-ended parallel universe concept. After the Crisis, the idea was they had one timeline, but eventually they established Hypertime, and all the while, had Elseworlds books. Elseworlds are one-shot stories that take place in parallel universes. They're a way to tell stories with heroes inspired by the mainstream continuity DC heroes without actually using said heroes, or to tell What-if? style stories. After Infinite Crisis, DC established a new status quo of there being 52 parallel Earths. Prior to The New 52, most DC books took place on New Earth, one of those 52 parallel Earths. I'm not familiar with what The New 52's multiverse status is.