  Previous work by the authors on parity-preserving fixed-length constrained
encoders is extended to the variable-length case. Parity-preserving
variable-length encoders are formally defined, and, to this end, Kraft
conditions are developed for the parity-preserving variable-length setting.
Then, a necessary and sufficient condition is presented for the existence of
deterministic parity-preserving variable-length encoders for a given
constraint. Examples are provided that show that there are coding ratios where
parity-preserving variable-length encoders exist, while fixed-length encoders
do not.
