Mormons believe heaven is multiple planets at its highest level, if you would classify them as Christians. I do not, for as a Christian I believe the very things that define my religion are the things on which I and mormons disagree.
Heaven cannot be another dimension because dimensions are simply things that can be measured like length and time. They are not places as science fiction has largely led people to believe. Heaven also cannot be a parallel reality, for such does not exist. This is a conflation of two terms: parallel universe and alternate reality. A parallel universe would be a sort of bubble, along with ours and other universes, within a larger multiverse. By definition, things that are parallel cannot intersect, so we could never get to heaven, nor could beings from heaven interact with us.
Alternate realities are the cause of higher dimensions. The dimensions we use on a daily basis are length, width, height, and time. Dimensions five and up, or even how many they are, have been largely debated and may not exist at all. Currently, time can only be measured in two directions, which we depict with a time line. Additional dimensions would allow time to be displayed on a coordinate graph rather than a single line. Every decision made by every person in time and also every random event in time could create alternate timelines from our reality (i.e. alternate realities). It can also be argued that certain events early enough in the formation of a universe would result in the formation not only of an alternate reality but of a separate place (parallel universe) within the multiverse. However, like I said, it is all a bunch of theoretical debate that cannot yet be supported by empirical evidence and is difficult to model mathematically.
As a Christian, I believe heaven means being with God. Hell is separation from God. I am therefore in heaven now, and heaven is in me, and heaven exists on earth where Christians are gathered. Heaven as a physical place and ultimate destination is the New Jerusalem, which will come into existence as part of the new heaven and new earth to replace the old when they are destroyed at Armageddon. It is not accurate to say heaven does not yet exist, however, because God and his angels must exist somewhere outside of time and space in order to be fully aware of them and are therefore just as much in the future as in the past and present.