Set the tabular inside a \makebox[\linewidth] box, similar to the suggestion in Centre an image, ignoring margins:
\documentclass{article}
\usepackage{amsmath,caption,threeparttable,booktabs}
\newcommand{\citet}[1]{#1}% Just for this example
\begin{document}
\begin{table}
\begin{threeparttable}
\renewcommand\TPTminimum{\linewidth}
\captionsetup{font=small}
\makebox[\linewidth]{\scriptsize%
\begin{tabular}{cccc}
\toprule
\textbf{Parameter} & \textbf{Value} & \textbf{Unity} & \textbf{Reference/Comment} \\
\midrule
$\tau_1$ & $10^{-2}$ & $\text{dia}^{-1}$ & \citet{Spratt} \\
$\tau_2$ & $10^{-3}$ & $\text{dia}^{-1}$ & $\tau_2 \; < \; \tau_1$ \\
%some more lines...
$\lambda$ & $4.16$ & $\text{dia}^{-1}$ & \citet{REF2}\tnote{2} \\
$a$ & $2 \times 10^{3}$ & mg & \citet{REF1}\tnote{1} \\
$b$ & $5 \times 10^{6}$ & mg & \citet{REF1}\tnote{1} \\
\bottomrule
\end{tabular}}
%NOTES:
\begin{tablenotes}[flushleft]\footnotesize
\item[1] I want it that way because my table isn't so big and the footnotes are getting weird the original way...
\item[2] \,Second Note!
\end{tablenotes}
\end{threeparttable}
\caption{Parameters...}
\end{table}
\end{document}
(Too long for a comment, will delete if this doesn't solve the problem.)
In the below code you'll find two versions of the table, one where it fits into a single column, one where it's wider. The table doesn't extend to the width of the page. But then, if you're submitting this to Elsevier they'll likely re-typeset everything anyway.
The lipsum package and \lipsum commands is only there to create dummy text, don't include them in your own document.
\documentclass[preprint,twocolumn,3p]{elsarticle}
\usepackage{threeparttable,array,booktabs}
\usepackage{lipsum}
\begin{document}
\lipsum[1] %dummy text
\begin{table}
\centering
\caption{Identified search terms for the above defined Research Questions.}
\label{t0}
\begin{threeparttable}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l >{\raggedright\arraybackslash}p{0.75\columnwidth}}
\toprule
RQ.~\# & Identified Terms\\
\midrule
RQ.~1 & SOA\textsuperscript{*}, cost estimation\textsuperscript{*}, effort estimation\textsuperscript{*} \\
RQ.~2 & approach\textsuperscript{*}, technique\textsuperscript{*}, SOA cost, SOA scope \\
RQ.~3 & classification\textsuperscript{*}, ranking, type \\
RQ.~4 & validation, evaluation, empirical study\textsuperscript{*} \\
RQ.~5 & future work\textsuperscript{*}, improvement \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\item [*] Note: These search terms are having synonyms or alternate spellings or abbreviations.
\end{tablenotes}
\end{threeparttable}
\end{table}
\lipsum[2]
\begin{table*}
\centering
\caption{Identified search terms for the above defined Research Questions.}
\label{t1}
\begin{threeparttable}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l l}
\toprule
RQ.~\# & Identified Terms\\
\midrule
RQ.~1 & SOA\textsuperscript{*}, cost estimation\textsuperscript{*}, effort estimation\textsuperscript{*} \\
RQ.~2 & approach\textsuperscript{*}, technique\textsuperscript{*}, SOA cost, SOA scope \\
RQ.~3 & classification\textsuperscript{*}, ranking, type \\
RQ.~4 & validation, evaluation, empirical study\textsuperscript{*} \\
RQ.~5 & future work\textsuperscript{*}, improvement \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\item [*] Note: These search terms are having synonyms or alternate spellings or abbreviations.
\end{tablenotes}
\end{threeparttable}
\end{table*}
\lipsum\lipsum
\end{document}
Best Answer
Please compare this two tables if they are different in their alignment. If that is the case, please check if your packages are all up to date. If they are, we have to compare versions. For this, add
\listfilesto the preamble and get the package versions from themain.logfile.