Super and subscripts with \DeclarePairedDelimiter

You need to get the \infty outside of the grouping. And that will involve redoing the options:

\documentclass[a4paper]{memoir}
\usepackage{mathtools,xparse}
\DeclarePairedDelimiter{\supnormX}{\lVert}{\rVert}
\DeclareDocumentCommand\supnorm{ s o m }{%
  \IfBooleanTF{#1}{% starred
    \supnormX*{#3}_\infty
  }{% not starred
    \IfNoValueTF{#2}{% no []
      \supnormX{#3}_\infty
    }{% data in []
      \supnormX[#2]{#3}_\infty
    }
  }
}
\begin{document}
\[
\supnorm{A}^2
\]
\end{document}

Here's a different implementation of \DeclarePairedDelimiter for which I use a different name.

\documentclass[a4paper]{memoir}
\usepackage{amsmath}
\usepackage{xparse}

\ExplSyntaxOn
\DeclareDocumentCommand{\xDeclarePairedDelimiter}{mmmO{}}
 {
  \NewDocumentCommand{#1}{sO{}m}
   {
    \IfBooleanTF{##1}
     {
      \egreg_paired_delimiter_expand:nnnn {#2}{#3}{##3}{#4}
     }
     {
      \egreg_paired_delimiter_fixed:nnnnn {##2}{#2}{#3}{##3}{#4}
     }
   }
 }
\cs_new_protected:Npn \egreg_paired_delimiter_expand:nnnn #1 #2 #3 #4
 {% Fix the spacing issue with \left and \right (D. Arsenau, P. Stephani and H. Oberdiek)
  \mathopen{}
  \mathclose\c_group_begin_token
   \left#1
   #3
   \group_insert_after:N \c_group_end_token
   \right#2
   \tl_if_empty:nF {#4} { \c_math_subscript_token {#4} }
 }
\cs_new_protected:Npn \egreg_paired_delimiter_fixed:nnnnn #1 #2 #3 #4 #5
 {
  \mathopen{#1#2}#4\mathclose{#1#3}
  \tl_if_empty:nF {#5} { \c_math_subscript_token {#5} }
 }
\ExplSyntaxOff

%% the final optional argument to \xDeclarePairedDelimiter
%% is a subscript to the right fence
\xDeclarePairedDelimiter{\supnormX}{\lVert}{\rVert}[\infty]

\begin{document}
\[
\supnormX{A}^2\quad
\supnormX[\big]{A}^2\quad
\supnormX[\Big]{A}^2\quad
\supnormX[\bigg]{A}^2\quad
\supnormX[\Bigg]{A}^2\quad
\supnormX*{\frac{A}{2}}^2
\]
\end{document}

---

Since daleif has shown interest in extending this idea, I add a way to define the macros with a key-value interface. The main (internal) functions remain the same.

\documentclass[a4paper]{memoir}
\usepackage{amsmath}
\usepackage{xparse}

\ExplSyntaxOn
\DeclareDocumentCommand{\KDeclarePairedDelimiter}{mm}
 {
  \__egreg_delimiter_clear_keys: % reset to the default
  \keys_set:nn { egreg/delimiters } { #2 }
  \use:x % we want to expand the values of the token variables set with the keys
   {
    \exp_not:n {\NewDocumentCommand{#1}{sO{}m} }
     {
      \exp_not:n { \IfBooleanTF{##1} }
       {
        \exp_not:N \egreg_paired_delimiter_expand:nnnn
         { \exp_not:V \l_egreg_delimiter_left_tl }
         { \exp_not:V \l_egreg_delimiter_right_tl }
         { \exp_not:n { ##3 } }
         { \exp_not:V \l_egreg_delimiter_subscript_tl }
       }
       {
        \exp_not:N \egreg_paired_delimiter_fixed:nnnnn 
         { \exp_not:n { ##2 } }
         { \exp_not:V \l_egreg_delimiter_left_tl }
         { \exp_not:V \l_egreg_delimiter_right_tl }
         { \exp_not:n { ##3 } }
         { \exp_not:V \l_egreg_delimiter_subscript_tl }
       }
     }
   }
 }

\keys_define:nn { egreg/delimiters }
 {
  left      .tl_set:N = \l_egreg_delimiter_left_tl,
  right     .tl_set:N = \l_egreg_delimiter_right_tl,
  subscript .tl_set:N = \l_egreg_delimiter_subscript_tl,
 }

\cs_new_protected:Npn \__egreg_delimiter_clear_keys:
 {
  \keys_set:nn { egreg/delimiters } { left=.,right=.,subscript={} }
 }

\cs_new_protected:Npn \egreg_paired_delimiter_expand:nnnn #1 #2 #3 #4
 {% Fix the spacing issue with \left and \right (D. Arsenau, P. Stephani and H. Oberdiek)
  \mathopen{}
  \mathclose\c_group_begin_token
   \left#1
   #3
   \group_insert_after:N \c_group_end_token
   \right#2
   \tl_if_empty:nF {#4} { \c_math_subscript_token {#4} }
 }
\cs_new_protected:Npn \egreg_paired_delimiter_fixed:nnnnn #1 #2 #3 #4 #5
 {
  \mathopen{#1#2}#4\mathclose{#1#3}
  \tl_if_empty:nF {#5} { \c_math_subscript_token {#5} }
 }
\ExplSyntaxOff

\KDeclarePairedDelimiter{\supnormX}{
  left=\lVert,
  right=\rVert,
  subscript=\infty
}

\begin{document}
\[
\supnormX{A}^2\quad
\supnormX[\big]{A}^2\quad
\supnormX[\Big]{A}^2\quad
\supnormX[\bigg]{A}^2\quad
\supnormX[\Bigg]{A}^2\quad
\supnormX*{\frac{A}{2}}^2
\]
\end{document}

Update:

Now with all the options provided by \DeclarePairedDelimiter! See the new definition (without using xparse):

\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}
\makeatletter
\newcommand{\@supnormstar}[1]{\norm*{#1}_\infty}
\newcommand{\@supnormnostar}[2][]{\norm[#1]{#2}_\infty}
\newcommand{\supnorm}{\@ifstar\@supnormstar\@supnormnostar}
\makeatother

For example, the following code:

\[\supnorm{\frac{\sqrt5-1}{2}}^2\]
\[\supnorm*{\frac{\sqrt5-1}{2}}^2\]
\[\supnorm[\big]{\frac{\sqrt5-1}{2}}^2\]

will give you: