This paper introduces a practical method for distributed lossy compression (Wyner-Ziv quantization) with side information available only at the decoder, where the side information is equal to the signal affected by background noise and additional impulse noise. At the core of the method is an LDPC-based lossless distributed (Slepian-Wolf) source code for q-ary alphabets, which is matched to the impulse probability and allows to remove the scalar-quantized impulse noise. Applications of this method to distributed compressed sensing of signals that differ in a sparse set of locations is also discussed, as well as some differences and similarities of variable- and fixed-length coding of sparse signals.