We investigate the ISM properties of 13 star-forming galaxies within the z ~ 2 COSMOS cluster. We show that the cluster members have [N II]/Hα and [O III]/Hβ emission-line ratios similar to z ~ 2 field galaxies, yet systematically different emission-line ratios (by ~0.17 dex) from the majority of local star-forming galaxies. We find no statistically significant difference in the [N II]/Hα and [O III]/Hβ line ratios or ISM pressures among the z ~ 2 cluster galaxies and field galaxies at the same redshift. We show that our cluster galaxies have significantly larger ionization parameters (by up to an order of magnitude) than local star-forming galaxies. We hypothesize that these high ionization parameters may be associated with large specific star formation rates (SFRs; i.e., a large SFR per unit stellar mass). If this hypothesis is correct, then this relationship would have important implications for the geometry and/or the mass of stars contained within individual star clusters as a function of redshift. ; 10 page(s)