We wish to study the extent and subparsec scale spatial structure of intervening quasar absorbers, mainly those involving neutral and molecular gas. We have selected quasar absorption systems with high spectral resolution and good S/N data, with some of their lines falling on quasar emission features. By investigating the consistency of absorption profiles seen for lines formed either against the quasar continuum source or on the much more extended emission line region (ELR), we can probe the extent and structure of the foreground absorber over the extent of the ELR (0.3-1 pc). The spatial covering analysis provides constraints on the transverse size of the absorber and thus is complementary to variability or photoionisation modelling studies. The methods we used to identify spatial covering or structure effects involve line profile fitting and curve of growth analysis.We have detected three absorbers with unambiguous non uniformity effects in neutral gas. For one extreme case, the FeI absorber at z_abs=0.45206 towards HE 0001-2340, we derive a coverage factor of the ELR of at most 0.10 and possibly very close to zero; this implies an absorber overall size no larger than 0.06 pc. For the z_abs=2.41837 CI absorber towards QSO J1439+1117, absorption is significantly stronger towards the ELR than towards the continuum source in several CI and CI* velocity components pointing to factors of about two spatial variations of their column densities and the presence of structures at the 100 au - 0.1 pc scale. The other systems with firm or possible effects can be described in terms of partial covering of the ELR, with coverage factors in the range 0.7 - 1. The overall results for cold, neutral absorbers imply a transverse extent of about five times or less the ELR size, which is consistent with other known constraints.
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