"Sparsing" is a combination of the words "sparse" and "parsing." In its simplest form it means simply incompletely parsing the data you have as input (or taking only sparse samples from it) - a common technique is quadratic sparsing, in which you take samples 1, 4, 9, 16, 25, 36, etc., throwing out the ones in between. In more refined forms it is often used with predictive algorithms to throw away data that is likely to be uninteresting (instead of just throwing out arbitrary data as with quadratic sparsing). It's a technique often used in physics applications to try to turn gigabytes of experimental data into something manageable.
In terms of LAME, I'm not 100% sure what it is being used to refer to. The entire psychoacoustic model would seem to be some sort of a predictive algorithm to do intelligent sparsing (throw out data that is unlikely to be audible). Searching google I also encountered use of the term to refer to zeroing out pairs of side-channel samples to reduce the content of the side channel in M/S stereo, which might just be one specific instance where the term is used:
http://minnie.tuhs.org/pipermail/mp3encode...May/002859.html
But generally almost everything to do with perceptual encoding could probably be described as "sparsing," though in the audio-coding community it might only be used to refer to one particular instance; someone more familiar with the community than I will have to answer how the term is generally used in the context of mp3-coding.