Reference to List Items in Theorem Environment with Label Type “Theorem”

This is an attempt based on your example. I used the theorem hooks defined by thmtools to override the format for thmlisti in the current environment:

\addtotheorempostheadhook[thm]{\crefalias{thmlisti}{thm}}
\addtotheorempostheadhook[lem]{\crefalias{thmlisti}{lem}}

I cannot promise that it won't break anything, for example if you use a thmlist without any surrounding theorem.

\documentclass[10pt,a4paper]{article}

% Input Type and AMS-Packages 
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amsfonts,amssymb,amsthm}

% Typography
\usepackage{enumitem}

\usepackage{letltxmacro}

\newlist{thmlist}{enumerate}{1}
\setlist[thmlist]{label=(\roman{thmlisti}), ref=\thethm.(\roman{thmlisti}),noitemsep}


% Math, Operators and Theorems
\usepackage{thmtools}

\declaretheorem[
name=Theorem,
%refname={theorem,theorems},        %Lower Case Versions of Theorem Type
Refname={Theorem,Theorems},
numberwithin=section]{thm}
\declaretheorem[
name=Lemma,
%refname={lemma,lemmas},
Refname={Lemma,Lemmas},
sibling=thm]{lem}

% References
\usepackage{nameref,hyperref}
\usepackage[capitalize]{cleveref}

\Crefname{thm}{Theorem}{Theorems}
\Crefname{lem}{Lemma}{Lemmas}

%%% NEW
\addtotheorempostheadhook[thm]{\crefalias{thmlisti}{thm}}
\addtotheorempostheadhook[lem]{\crefalias{thmlisti}{lem}}

\begin{document}

\begin{thm}\label{thm:A}
  \begin{thmlist}
    \item Statment 1\label{thm:A1}
    \item Statment 2\label[thm]{thm:A2}
  \end{thmlist}
\end{thm}

\begin{lem}\label{thm:B}
  \begin{thmlist}
    \item Statment 1\label{thm:B1}
    \item Statment 2\label[lem]{thm:B2}
  \end{thmlist}
\end{lem}

\begin{description}
  \item[1:] That's how I refer to theorems and their like: \cref{thm:A} and \cref{thm:B}.
  \item[2:] That's how I refer to parts of theorems but without the label type: \ref{thm:A1} and \ref{thm:B1}.
  \item[3:] This happens when using \verb+\autoref{•}:+ \autoref{thm:A1} and \autoref{thm:B1}.
  \item[4:] This happens when using \verb+\cref{•}:+ \cref{thm:A1} and \cref{thm:B1} (now working)
  \item[5:] My current work around: \cref{thm:A2} and \cref{thm:B2}.
\end{description}
\end{document}

Here's a solution.

Define a new list lemlist

\newlist{lemlist}{enumerate}{1}
\setlist[lemlist]{label=(\roman{lemlisti}), ref=\thethm.(\roman{lemlisti}),noitemsep}

and use this one inside lem.

Then, declare

\Crefname{thmlisti}{Theorem}{Theorems}
\Crefname{lemlisti}{Lemma}{Lemmas}

and you will be able to avoid the optional argument for \labels.

MWE:

\documentclass[10pt,a4paper]{article}

% Input Type and AMS-Packages
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amsfonts,amssymb,amsthm}

% Typography
\usepackage{enumitem}

\newlist{thmlist}{enumerate}{1}
\setlist[thmlist]{label=(\roman{thmlisti}), ref=\thethm.(\roman{thmlisti}),noitemsep}
\newlist{lemlist}{enumerate}{1}
\setlist[lemlist]{label=(\roman{lemlisti}), ref=\thethm.(\roman{lemlisti}),noitemsep}

% Math, Operators and Theorems
\usepackage{thmtools}

\declaretheorem[
    name=Theorem,
    %refname={theorem,theorems},        %Lower Case Versions of Theorem Type
    Refname={Theorem,Theorems},
    numberwithin=section]{thm}
\declaretheorem[
    name=Lemma,
    %refname={lemma,lemmas},
    Refname={Lemma,Lemmas},
    sibling=thm]{lem}

% References
\usepackage{nameref,hyperref}
\usepackage[capitalize]{cleveref}

\Crefname{thm}{Theorem}{Theorems}
\Crefname{lem}{Lemma}{Lemmas}
\Crefname{thmlisti}{Theorem}{Theorems}
\Crefname{lemlisti}{Lemma}{Lemmas}

\begin{document}

\begin{thm}\label{thm:A}
\begin{thmlist}
\item Statment 1\label{thm:A1}
\item Statment 2\label{thm:A2}
\end{thmlist}
\end{thm}

\begin{lem}\label{thm:B}
\begin{lemlist}
\item Statment 1\label{thm:B1}
\item Statment 2\label{thm:B2}
\end{lemlist}
\end{lem}

\begin{description}
\item[1:] That's how I refer to theorems and their like: \cref{thm:A} and \cref{thm:B}.
\item[2:] That's how I refer to parts of theorems but without the label type: \ref{thm:A1} and \ref{thm:B1}.
\item[3:] This happens when using \verb+\autoref{•}:+ \autoref{thm:A1} and \autoref{thm:B1}.
\item[4:] This happens when using \verb+\cref{•}:+ \cref{thm:A1} and \cref{thm:B1}.
\item[5:] My current work around: \cref{thm:A2} and \cref{thm:B2}.
\end{description}
\end{document} 

Output:

I was able to improve the great answer by @T.Verron such that \ref will return the plain counter of the list-environments while \cref will return the complete label compromising type, number of theorem and counter of the list (see my comment on @T.Verron's answer).

To do so I set up two new counters

\newcounter{listthm}
\newcounter{listlem}

whose names correspond to the theorem environment I want to use them in

\Crefname{thm}{Theorem}{Theorems}
\Crefname{lem}{Lemma}{Lemmas}
\Crefname{listthm}{Theorem}{Theorems}
\Crefname{listlem}{Lemma}{Lemmas}

In the theorem's headposthook the style of thmlistiis now set to listthmor listlemrespectively.

\addtotheorempostheadhook[thm]{\crefalias{thmlisti}{listthm}}
\addtotheorempostheadhook[lem]{\crefalias{thmlisti}{listlem}}

The critical step is that I am now able to tinker with the label format of listthmand listlemwithout affecting the format of thmand lem solely making use of commands provided by the cref-package

\creflabelformat{listthm}{#2\thethm.#1#3}
\creflabelformat{listlem}{#2\thethm.#1#3}

Now I can avoid setting the ref-style of thmlistiusing the enumitem-package which caused the \ref-command to return the counter of the theorem as well.

For full details see my example below.

\documentclass[10pt,a4paper]{article}

% Input Type and AMS-Packages 
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amsfonts,amssymb,amsthm}

% Typography
\usepackage{enumitem}

\newlist{thmlist}{enumerate}{1}
\setlist[thmlist]{label=(\roman{thmlisti}),noitemsep}

% Math, Operators and Theorems
\usepackage{thmtools}

\declaretheorem[
    name=Theorem,
    numberwithin=section]{thm}
\declaretheorem[
    name=Lemma,
    sibling=thm]{lem}

% References
\usepackage{hyperref}
\usepackage[capitalize]{cleveref}

\newcounter{listthm}
\newcounter{listlem}


\Crefname{thm}{Theorem}{Theorems}
\Crefname{lem}{Lemma}{Lemmas}
\Crefname{listthm}{Theorem}{Theorems}
\Crefname{listlem}{Lemma}{Lemmas}


\addtotheorempostheadhook[thm]{\crefalias{thmlisti}{listthm}}
\addtotheorempostheadhook[lem]{\crefalias{thmlisti}{listlem}}

\creflabelformat{listthm}{#2\thethm.#1#3}
\creflabelformat{listlem}{#2\thethm.#1#3}

\begin{document}

\begin{thm}\label{thm:A}
\begin{thmlist}
\item Statment 1\label{thm:A1}
\item Statment 2\label{thm:A2}
\end{thmlist}
\end{thm}
\begin{lem}\label{thm:B}
\begin{thmlist}
\item Statment 1\label{thm:B1}
\item Statment 2\label{thm:B2}
\end{thmlist}
\end{lem}


\begin{enumerate}
\item That's how I refer to theorems and their like: \cref{thm:A} and \cref{thm:B}.
\item Referring to parts of theorems works like a charm with \verb+\cref{•}:+ \cref{thm:A1} and \cref{thm:B1}.
\item If I use \verb+\ref{•}+ to refer to a part of a theorem \LaTeX{} returns \ref{thm:A1} and \ref{thm:B2}.
\end{enumerate}
\end{document}