Start inparaenum-numeration with another letter than "a)"

You can set the number by hand; since it's a second level list, you have to set enumii.

\documentclass[14pt,a4paper,headlines=6,headinclude=true]{scrartcl}

\usepackage{amsmath,amssymb,stmaryrd}       
\usepackage{paralist}                                       
\usepackage{tabto}

\begin{document}

\begin{enumerate}
    \item \textbf{Working with logarithms}: 

    \NumTabs{3}
    \begin{inparaenum}[a)]
    Calculate the following

    \item $\ln 1 = $ 
    \tab \item $\ln e = $ 
    \tab \item $\ln \frac{1}{e} = $
    \end{inparaenum} 

    Express the following in terms of $\ln 2$: 

    \NumTabs{3}
    \begin{inparaenum}[a)]\setcounter{enumii}{3}
    \item $\ln 4$ 
    \tab \item $\ln \sqrt[3]{2^5}$ 
    \tab \item $\ln \frac{1}{16} $
    \end{inparaenum} 
\end{enumerate}

\end{document}

You can use, instead, enumitem:

\documentclass[14pt,a4paper,headlines=6,headinclude=true]{scrartcl}

\usepackage{amsmath,amssymb,stmaryrd}
\usepackage[inline]{enumitem}
\usepackage{tabto}

\begin{document}

\begin{enumerate}
\NumTabs{3}
\item \textbf{Working with logarithms}:

Calculate the following\\[\medskipamount]
  \begin{enumerate*}[label=\alph*),itemjoin=\tab,before={}]
  \item $\ln 1 = $ 
  \item $\ln e = $ 
  \item $\ln \frac{1}{e} = $
  \end{enumerate*}

Express the following in terms of $\ln 2$:\\[\medskipamount]
  \begin{enumerate*}[label=\alph*),itemjoin=\tab,resume]
  \item $\ln 4$ 
  \item $\ln \sqrt[3]{2^5}$ 
  \item $\ln \frac{1}{16} $
  \end{enumerate*} 
\end{enumerate}

\end{document}

Another version, using my assoccnt package, without the need of remembering the counter value:

The contenumii counter is used to store the total value of the enumii counter automatically and stored back after inparaenum.

Caveat: It will be incremented each time when another second level enumeration is used

Use the \SuspendCounters feature from the assoccnt package then

\documentclass[14pt,a4paper,headlines=6,headinclude=true]{scrartcl}

\usepackage{amsmath,amssymb,stmaryrd}       
\usepackage{paralist}                                       
\usepackage{tabto}

\usepackage{assoccnt}



\newcounter{contenumii}
\DeclareAssociatedCounters{enumii}{contenumii}


\begin{document}

\begin{enumerate}
    \item \textbf{Working with logarithms}: 

    \NumTabs{3}
    \begin{inparaenum}[a)]
    Calculate the following

    \item $\ln 1 = $ 
    \tab \item $\ln e = $ 
    \tab \item $\ln \frac{1}{e} = $
    \end{inparaenum} 

    Express the following in terms of $\ln 2$: 

    \NumTabs{3}
    \begin{inparaenum}[a)]\setcounter{enumii}{\number\value{contenumii}}
    \item $\ln 4$ 
    \tab \item $\ln \sqrt[3]{2^5}$ 
    \tab \item $\ln \frac{1}{16} $
    \end{inparaenum} 

\end{enumerate}

\end{document}

Other options: enumitem package, with the resume option, but this is not as compact as paralist (initially, can be changed, perhaps)