What is a parallel mode? How are the parallel modes used in composition and/or improvisation?
Two modes are parallel if they share the same tonic. That is, D Major, D Minor, D Dorian, and D Mixolydian are all parallel modes. Using a parallel mode will cause a chromatic alteration to your usual key signature. For example, Dorian uses #6 and Phrygian uses b2 (when compared to a minor key or Aeolian mode), while Mixolydian uses b7 and Lydian uses #4 (compared to a major key, or Ionian mode).
By way of contrast, two modes are relative if they share the same key signature. D Major, B Minor, A Mixolydian, and E Dorian are all relative modes. Using a relative mode will not generally cause chromatic alterations, but will cause the tonic to shift to a different note.
In composition (or improvisation) parallel modes allow the use of borrowed chords, via modal mixture, to add color to piece. One example would be the use of the minor iv chord in place of IV, in the major key. Another example might be bIII, bVI, and bVII chords which are common in certain styles of rock.
The most common form of parallel modes is between parallel major and minor keys. Some pieces in a minor key will borrow the parallel major I chord at the very end, a technique known as a Picardy Third. In some minor key symphonies, its possible to find the entire final movement is in the parallel major (such as Beethoven's 5th). Some Baroque composers such as Vivaldi would occasionally repeat a major theme in a parallel minor to give it a new dimension. Even the ascending melodic minor scale can be thought of as a form of modal mixture, borrowing its notes from the parallel major.
In short, they give you new harmonic "colors" to use, without forcing you to change your tonic.
To expand a little on Caleb's great answer, let's take C major. No key signature, as all notes are 'white' - no # or b. Parallels are - C minor with 3 b. C Dorian with 2 b.C Phrygian with 4b, C Lydian with 1 #, C Mixolydian with 1b, C Aeolian with 3b (as minor), C Locrian with 5b. This example is the 'ordinary' modes, showing that home or the root of C can involve loads of other notes which can manifest themselves when the tune moves to a parallel mode, still 'at home' in C something.
Same roots but different notes. C Ionian, C Dorian, C Phrygian, etc