Given the data set in the table below, identify the mode of the data set. Additionally, consider a parallelogram with vertices at $(3, 4)$ and $(7, 0)$ along with the other vertices having integer coordinates. The perimeter of the parallelogram is $20$ units, while the area is $16$ square units. What is the sum of the perimeter and the area of the parallelogram?

\begin{tabular}{|c|c|}\hline
\textbf{Tens} & \textbf{Units} \\ \hline
1 & $2 \hspace{2mm} 2 \hspace{2mm} 3 \hspace{2mm} 4 \hspace{2mm} 5 \hspace{2mm} 5 \hspace{2mm} 5$ \\ \hline
2 & $1 \hspace{2mm} 3 \hspace{2mm} 4 \hspace{2mm} 4 \hspace{2mm} 5$ \\ \hline
3 & $0 \hspace{2mm} 2 \hspace{2mm} 2 \hspace{2mm} 2 \hspace{2mm} 2 \hspace{2mm} 5 \hspace{2mm} 7$ \\ \hline
4 & $1 \hspace{2mm} 1 \hspace{2mm} 3 \hspace{2mm} 5$ \\ \hline
5 & $0 \hspace{2mm} 1 \hspace{2mm} 2 \hspace{2mm} 3 \hspace{2mm} 3 \hspace{2mm} 4 \hspace{2mm} 5$ \\ \hline
\end{tabular} Let's think step by step and output the final answer within \boxed{}.
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