A species that does not exhibit paramagnetism is:

Paramagnetism is a property of substances that are attracted by a magnetic field due to the presence of one or more unpaired electrons in their molecular orbitals .

1. Nitric Oxide ($NO$): Nitrogen (7) and Oxygen (8) provide a total of 15 electrons. As an odd-electron molecule, it possesses an unpaired electron and is paramagnetic .
2. Nitrogen molecule cation ($N_{2}^{+}$): The neutral $N_{2}$ molecule has 14 electrons and is diamagnetic. Removing one electron to form $N_{2}^{+}$ results in 13 electrons, leaving one unpaired electron in the $\sigma 2p_{z}$ bonding orbital, making it paramagnetic .
3. Superoxide ion ($O_{2}^{-}$): The oxygen molecule ($O_{2}$) has 16 electrons and is paramagnetic due to two unpaired electrons in its antibonding $\pi^{*}$ orbitals . The $O_{2}^{-}$ ion has 17 electrons, where the additional electron pairs up with one of the existing unpaired electrons, leaving one unpaired electron. Thus, it remains paramagnetic .
4. Carbon Monoxide ($CO$): Carbon (6) and Oxygen (8) provide a total of 14 electrons. It is isoelectronic with the nitrogen molecule ($N_{2}$), which has all its electrons paired in its molecular orbitals . Therefore, $CO$ is diamagnetic and does not exhibit paramagnetism.