First one is a sub section and hence you can load titlesec package and use its titlespacing* macro
\usepackage{titlesec}
\titlespacing*{\subsection}{0pt}{0pt}{-10pt}
The third argument is the space after (below) the title which I have made -10pt. Adjust as you wish.
For the second space, you can apply the original \parskip value (i. e., 0pt plus 1pt) only for the enumerate by
before=\setlength{\parskip}{0pt plus 1pt},align=left,....
in the options.
\documentclass[11pt]{book}
\usepackage{enumitem}
\setlist{nolistsep}
\usepackage{amsmath,amsfonts,amssymb,amsthm, bm}
\parskip = \baselineskip
\usepackage{titlesec}
\titlespacing*{\subsection}{0pt}{0pt}{-10pt}
\begin{document}
\chapter{Problems from Chapter 9}
\section{Problem 9-3}\index{Problem 9-3}
\subsection{Problem 9-3}
(Sorenson, 1980, Exercise 3-15, p. 130). Suppose $\theta_{1}$ and $\theta_{2}$ are unbiased estimators of $\theta$ with var$(\theta_{1}) = \sigma_{1}^{2}$ and var$(\theta_{2}) = \sigma_{2}^{2}$. Let $\theta_{3} = \alpha\theta_{1} + (1-\alpha)\theta_{2}$.
%
\begin{enumerate}[before=\setlength{\parskip}{0pt plus 1pt},align=left,style=nextline,leftmargin=*, labelsep=\parindent,label=\textbf{(\alph*)}]
\item Prove that $\theta_{3}$ is unbiased.
\item Assume that $\theta_{1}$ and $\theta_{2}$ are statically independent, and find the measured squared error of $\theta_{3}$.
\item What choice of $\alpha$ minimizes the mean-squared error?
\end{enumerate}
\end{document}
Instead of adjusting \parskip yourself, you may use parskip package which gives better spacing.
\documentclass[11pt]{book}
\usepackage{enumitem}
\setlist{nolistsep}
\usepackage{amsmath,amsfonts,amssymb,amsthm, bm}
\usepackage[parfill]{parskip}
\begin{document}
\chapter{Problems from Chapter 9}
\section{Problem 9-3}\index{Problem 9-3}
\subsection{Problem 9-3}
(Sorenson, 1980, Exercise 3-15, p. 130). Suppose $\theta_{1}$ and $\theta_{2}$ are unbiased estimators of $\theta$ with var$(\theta_{1}) = \sigma_{1}^{2}$ and var$(\theta_{2}) = \sigma_{2}^{2}$. Let $\theta_{3} = \alpha\theta_{1} + (1-\alpha)\theta_{2}$.
%
\begin{enumerate}[align=left,style=nextline,leftmargin=*, labelsep=\parindent,label=\textbf{(\alph*)}]
\item Prove that $\theta_{3}$ is unbiased.
\item Assume that $\theta_{1}$ and $\theta_{2}$ are statically independent, and find the measured squared error of $\theta_{3}$.
\item What choice of $\alpha$ minimizes the mean-squared error?
\end{enumerate}
\end{document}