Parallels

To travel to Parallel Dimensions, think of them not as different destinations, but as the same destination. Like turning the channel on a television, you need only change the frequency of matter in a given area. If you were to build a machine, it can be as large or as small as you'd like, it must be capable of manipulating the most basic elements of the universe in an incredibly energetic fashion.
The power levels required would be astronomical but entirely possible. Likely on a similar level as any star like our own sun, but said energy would be released within seconds and regulated to an incredibly finite degree.
Moving from one dimension to another would require the assistance of a targeting system that functions both inside and outside of Space/Time as we know it. Likely one involving some sort of computer system built of both Matter and Antimatter. Existing in our Space/Time and not existing all at once. With this system, accompanied by a secondary system to manipulate the physical frequency of the object to be traveled, you could simultaneously establish your current location in one dimension, target that same location in another dimension, and chart a basic path through the barriers between the two.
While it is very possible, you must also take into consideration the physical forces at work during such a journey. Moving dimensions would be incredibly traumatic to the body, and any explorer would require a survival suit capable of resisting a journey which could tear apart a planet or star in moments.
Once the basic issues of; power, targeting, and survival are solved, you may move to any dimension, as easily as turning a dial.