How can I vertically center forest trees?

As defined, there is only one box and no other reference for alignment hence all trees are top-aligned. Choosing to put each tree on separate boxes and choosing the middle alignment, M the vertical center (equal height and depth), we obtain the desired result. Something like:

\begin{adjustbox}{valign=M} <content> \end{adjustbox}\qquad\begin{adjustbox}{valign=M} <content> \end{adjustbox}\qquad\begin{adjustbox}{valign=M} <content> \end{adjustbox}

Here is mwe:

\documentclass[a4size]{article}
\usepackage{microtype}
\usepackage[T1]{fontenc}
\usepackage{adjustbox}
% forest
\usepackage{forest}
% parsing tree
\forestset{
  declare toks={wff}{},
  declare toks={connective}{},
  declare dimen register={parsing tree sep},
  parsing tree sep=7.5pt,
  parsing tree/.style={
    % append the current root to a new phantom root.
    for root'={
      replace by={[,phantom,append=!last dynamic node]}
    },
    for tree={
      math content,
      parent anchor=children,
      child anchor=parent,
      inner sep=0pt,
      if n children=1{!first.before packing=calign with current edge}{},
      delay={
        content=\circ,
        insert before/.wrap pgfmath arg={
          [##1,no edge,math content,anchor=base east, before computing xy={
            s/.pgfmath={s("!n")-parsing_tree_sep}}]
        }{wff},
        if connective={}{connective/.option=wff}{},
        insert after/.wrap pgfmath arg={
          [##1,no edge,math content,anchor=base west, before computing xy={
            s/.pgfmath={s("!p")+parsing_tree_sep}}]
        }{connective}
      }
    }
  },
  default preamble={parsing tree,baseline}
}
\begin{document}
\begin{center}
  \fbox{
    \begin{adjustbox}{valign=M}
      \begin{forest}
        [,wff=1,connective=\chi]
      \end{forest}
    \end{adjustbox}
      \qquad
    \begin{adjustbox}{valign=M}
      \begin{forest}
        [,wff=m+3,connective=\neg [,wff=m,connective=\chi]]
      \end{forest}
    \end{adjustbox}
      \qquad
    \begin{adjustbox}{valign=M}
      \begin{forest}
        [,wff=m+n+3,connective=\neg [,wff=m,connective=\chi]
        [,wff=n,connective=\chi]]
      \end{forest}
    \end{adjustbox}
  }
\end{center}
\end{document}

Other methods involve using a table format. Something like:

\begin{tabular}{m{0.25\textwidth}m{0.25\textwidth}m{0.25\textwidth}@{}m{0pt}@{}}
  \begin{forest}
    [,wff=1,connective=\chi]
  \end{forest}
  &
  \begin{forest}
    [,wff=m+3,connective=\neg [,wff=m,connective=\chi]]
  \end{forest}
  &
  \begin{forest}
    [,wff=m+n+3,connective=\neg [,wff=m,connective=\chi]
    [,wff=n,connective=\chi]]
  \end{forest}
  &
\end{tabular}

You do not need different boxes etc. You just need to remove baseline and use TikZ's baseline option to centre the trees vertically. For example,

\documentclass[]{article}
\usepackage{microtype}
\usepackage[T1]{fontenc}
\usepackage{adjustbox}
% forest
\usepackage{forest}
% parsing tree
\forestset{
  declare toks={wff}{},
  declare toks={connective}{},
  declare dimen register={parsing tree sep},
  parsing tree sep=7.5pt,
  parsing tree/.style={
    % append the current root to a new phantom root.
    for root'={
      replace by={[,phantom,append=!last dynamic node]}
    },
    for tree={
      math content,
      parent anchor=children,
      child anchor=parent,
      inner sep=0pt,
      if n children=1{!first.before packing=calign with current edge}{},
      delay={
        content=\circ,
        insert before/.wrap pgfmath arg={
          [##1,no edge,math content,anchor=base east, before computing xy={
            s/.pgfmath={s("!n")-parsing_tree_sep}}]
        }{wff},
        if connective={}{connective/.option=wff}{},
        insert after/.wrap pgfmath arg={
          [##1,no edge,math content,anchor=base west, before computing xy={
            s/.pgfmath={s("!p")+parsing_tree_sep}}]
        }{connective}
      }
    },
    /tikz/baseline=(current bounding box.center)
  },
  default preamble={parsing tree}
}
\begin{document}
\begin{center}
  \fbox{
    \begin{adjustbox}{valign=M}
      \begin{forest}
        [,wff=1,connective=\chi]
      \end{forest}
      \qquad
      \begin{forest}
        [,wff=m+3,connective=\neg [,wff=m,connective=\chi]]
      \end{forest}
      \qquad
      \begin{forest}
        [,wff=m+n+3,connective=\neg [,wff=m,connective=\chi]
        [,wff=n,connective=\chi]]
      \end{forest}
    \end{adjustbox}
  }
\end{center}
\end{document}

Note that a4size is not a valid option so I've removed it as it makes no difference. Perhaps you meant a4paper? In that case, you should be sure to load something like geometry with a standard class.