Step 1: Concept
In metal carbonyls, the oxidation state of the central metal atom is generally zero because $CO$ is a neutral ligand.
Step 2: Analysis
In $Fe_{3}[Fe(CN)_{6}]_{2}$, Iron exists in $+2$ and $+3$ states.
In $Mn_{2}(CO)_{10}$, Manganese is bonded to neutral carbonyl ligands, resulting in an oxidation state of 0.
In $K[PtCl_{3}(C_{2}H_{4})]$ (Zeise's salt), Pt is in $+2$ state.
In $[Co(NH_{3})_{5}Br]_{2}SO_{4}$, Cobalt is in $+3$ state.
Step 3: Conclusion
Hence, the lowest oxidation state (0) is found in $Mn_{2}(CO)_{10}$.
Final Answer:(B)