1. Use Molecular Orbital (MO) theory to determine paramagnetism
   A molecule is paramagnetic if it has one or more unpaired electrons in its molecular orbitals.

2. Check each molecule

(A) $\mathrm{O_2}$

For $\mathrm{O_2}$, the MO electronic configuration contains two electrons in the degenerate antibonding orbitals $\pi^*_{2p_x}$ and $\pi^*_{2p_y}$, each singly occupied:

So, $\mathrm{O_2}$ has 2 unpaired electrons and is paramagnetic.

(B) $\mathrm{N_2}$

$\mathrm{N_2}$ has all electrons paired in bonding molecular orbitals. Hence it is diamagnetic.

(C) $\mathrm{F_2}$

$\mathrm{F_2}$ has all electrons paired, including in the antibonding orbitals. Hence it is diamagnetic.

(D) $\mathrm{S_2}$

$\mathrm{S_2}$ is analogous to $\mathrm{O_2}$ in valence-shell MO filling. It also has two unpaired electrons in $\pi^*$ antibonding orbitals. Hence $\mathrm{S_2}$ is paramagnetic.

(E) $\mathrm{Cl_2}$

$\mathrm{Cl_2}$ has all electrons paired, so it is diamagnetic.

3. Paramagnetic molecules Therefore, the molecules showing paramagnetism are:

So the correct option is:

4. Comparison with stored correct answer
   Stored correct answer = $\boxed{\text{C}}$
   Derived answer = $\boxed{\text{C}}$

They match.