Parabolic subgroups appear in the study of algebraic groups , BN-pairs and finite groups of Lie type...etc. The term parabolic is geometric and although there is a geometric approach for studying algebraic groups, it is not clear to me the connection of parabolic subgroups to "parabola". Any clew will be appreciated.
There seems to be some ambiguity in the history of this. Math overflow has had a good discussion on this.
http://mathoverflow.net/questions/24960/why-are-parabolic-subgroups-called-parabolic-subgroups
This is a good question.
To answer this question, we need to start with a connected linear algebraic group G. Connectedness is equivalent to irreducibility for a linear algebraic group.
The set of closed solvable subgroups of G partially ordered with inclusion $\subset$ contains maximal elements. The maximal element is called a Borel subgroup of G. If P is a closed subgroup of G, then G/B is a projective variety, if and only if P contains a Borel subgroup. Such a subgroup is called a parabolic subgroup. The name parabolic subgroup stems from the fact that G/B is a projective variety. For more about this, see
F. Murnaghan, Linear algebraic groups, Clay Math. Proc. 4, 2005:
http://www.math.toronto.edu/murnaghan/courses/algp.pdf
For more about the connection between the name parabolic and projectivity, see page 89 in
E.Z. Goren, Algebraic groups, part V, McGill University:
http://www.math.mcgill.ca/goren/MATH722.2010/AlgebraicGroups.5.pdf
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