  We classify two-dimensional local conformal nets with parity symmetry and
central charge less than 1, up to isomorphism. The maximal ones are in a
bijective correspondence with the pairs of A-D-E Dynkin diagrams with the
difference of their Coxeter numbers equal to 1. In our previous classification
of one-dimensional local conformal nets, Dynkin diagrams D_{2n+1} and E_7 do
not appear, but now they do appear in this classification of two-dimensional
local conformal nets. Such nets are also characterized as two-dimensional local
conformal nets with mu-index equal to 1 and central charge less than 1. Our
main tool, in addition to our previous classification results for
one-dimensional nets, is 2-cohomology vanishing for certain tensor categories
related to the Virasoro tensor categories with central charge less than 1.
