How to cite author and date, but only date in brackets with biblatex?

The biblatex package's \textcite directive is what you're looking for. (This directive is listed in the middle column of the first page of the "Biblatex Cheat Sheet" document you mention in your query.)

\RequirePackage{filecontents}
\begin{filecontents}{references.bib}
@misc{acharya_analysis_2002,
    title = "An Analysis of Apples",
    author = "Acharya, Alice and Benson, Bob and Carrey, Catalina and Duvet, Dorian",
    date = 2002
}
\end{filecontents}

\documentclass{article}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage[backend=biber,style=chicago-authordate]{biblatex}
\addbibresource{references.bib} 

\begin{document}
\textcite{acharya_analysis_2002} found that apples can come in a variety of colors.
\printbibliography
\end{document}

The best way to get Author (Year) citations in a biblatex author-year style is to use

\textcite

A manually-defined command like \newcommand{\citedateparen}[1]{\citeauthor{#1} (\citedate{#1})} has several drawbacks:

1. It does not handle multiple citations particularly well. Try \citedateparen{sigfridsson,worman}.
2. In this simple form it does not handle pre- and postnotes (pages etc.) at all.
3. It can mess up citation and ibidem tracking (this depends on the style and its settings).

The correct way to define a new citation command that does all these things properly is via \DeclareCiteCommand. But this is not required here, since the desired command already exists.

---

Note that the styles of biblatex-chicago should usually be loaded via the wrapper package biblatex-chicago and not via biblatex. Additionally, the MWE threw errors because one can't easily combine the style=alphabetic (or bibstyle=alphabetic) with citestyle=chicago-authordate due to the internal structure of chicago-authordate. (It also looks weird.)

\documentclass{article}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}

\usepackage[
  authordate,
  backend=biber,
]{biblatex-chicago}

\addbibresource{biblatex-examples.bib} 

\begin{document}
\textcite{sigfridsson} found that apples can come in a variety of colors.

\printbibliography
\end{document}