Hi John,

A couple of things. First, you have included the knitr and
short-inset-names modules in the document, which as far as I can tell are
not used for this minimal example. I removed them since I'm not yet set up
to work with those. This shouldn't cause any problems, but I thought I'd
mention it in case.

As a disclaimer, I am not exactly sure what I'm doing, but when I opened up
the apa.cls file and searched for paragraph and subparagraph, I found:
\renewcommand{\paragraph}{\@startsection{paragraph}{4}{\parindent}%
{0\baselineskip \@plus 0.2ex \@minus 0.2ex}%
{-1em}%

\renewcommand{\subparagraph}[1]{\@startsection{subparagraph}{5}{1em}%
{0\baselineskip \@plus 0.2ex \@minus 0.2ex}%
{-\z@\relax}%

{\normalfont\normalsize\itshape\hspace{\parindent}{#1}\textit{.}}{\relax}}

The thing is, with the document you sent, the paragraph part is working,
but the subparagraph part is not. So, if we use a little trial-and-error to
put them together, I end up with the following, which seems to work if you
paste it into your preamble (under Document->Settings->LaTeX Preamble).

\renewcommand{\subparagraph}{\@startsection{subparagraph}{5}{\parindent}%

{0\baselineskip \@plus 0.2ex \@minus 0.2ex}%

{-1em}%