Why is there a trace line in this tikz graph?

Big thanks to Sigur for explaining the question to me! The issue is that the paths do not have the appropriate orientation. So I had to reorder the sequences and reverse one to get

\documentclass[10pt,multi=False,border=5pt,tikz,class=scrartcl]{standalone}
\usepackage{xcolor}

\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\usetikzlibrary{arrows.meta}
\usetikzlibrary{patterns}
\usepgfplotslibrary{fillbetween}
\begin{document}
\begin{tikzpicture}
  \begin{axis}[grid=both, axis equal=false, ymin=-1, ymax=6, xmin=-30, xmax=30,
    minor tick num=1,axis lines = middle,
    label style={font=\small,at={(ticklabel cs:1.1)}},
    tick label style={font=\footnotesize}]
    \addplot [thick,dashed,samples=50, domain=-30:0,name path=p1] {sqrt(abs(x))};
    \addplot [thick,samples=50, domain=0:30,name path=p2] {sqrt(abs(x))};
    \addplot [thick,dashed,name path=p3] coordinates {(-25,5) (25,5)};
    \path [name path=left,  intersection segments={of=p1 and p3,sequence={R1[reverse] -- L2}}];
    \path [name path=right, 
    intersection segments={of=p2 and p3,sequence={L1 -- R2}}];
    \addplot [pattern=north east lines,pattern color=blue,opacity=.8] fill between [
        of=left and right,reverse=false];
    \draw [fill=gray,opacity=.5] (0,3.5) ellipse [x radius=12.25, y radius=.1];
    \addplot+ [only marks,mark=*] coordinates { (-25,5) (25,5)}
        node [pos=0,above right] {\footnotesize\textcolor{blue}{$B=(-25,5)$}}
        node [pos=1,above left] {\footnotesize\textcolor{blue}{$A=(25,5)$}};
    \node [right] at (11,3) {$r=x=y^2$};
  \end{axis}
\end{tikzpicture}
\end{document}

How can one debug this? Just draw these paths with arrows.

\documentclass[10pt,multi=False,border=5pt,tikz,class=scrartcl]{standalone}
\usepackage{xcolor}

\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\usetikzlibrary{arrows.meta}
\usetikzlibrary{patterns}
\usepgfplotslibrary{fillbetween}
\begin{document}
\begin{tikzpicture}
  \begin{axis}[grid=both, axis equal=false, ymin=-1, ymax=6, xmin=-30, xmax=30,
    minor tick num=1,axis lines = middle,
    label style={font=\small,at={(ticklabel cs:1.1)}},
    tick label style={font=\footnotesize}]
    \addplot [thick,dashed,samples=50, domain=-30:0,name path=p1] {sqrt(abs(x))};
    \addplot [thick,samples=50, domain=0:30,name path=p2] {sqrt(abs(x))};
    \addplot [thick,dashed,name path=p3] coordinates {(-25,5) (25,5)};
    \path [name path=left, thick,draw=blue,-latex,
    intersection segments={of=p1 and p3,sequence=L2 -- R1}];
    \path [name path=right, thick,draw=red,-latex,
    intersection segments={of=p2 and p3,sequence=L1 -- R2}];
    \addplot [pattern=north east lines,pattern color=blue,opacity=.8] fill between [
        of=left and right,reverse=false];
    \draw [fill=gray,opacity=.5] (0,3.5) ellipse [x radius=12.25, y radius=.1];
    \addplot+ [only marks,mark=*] coordinates { (-25,5) (25,5)}
        node [pos=0,above right] {\footnotesize\textcolor{blue}{$B=(-25,5)$}}
        node [pos=1,above left] {\footnotesize\textcolor{blue}{$A=(25,5)$}};
    \node [right] at (11,3) {$r=x=y^2$};
  \end{axis}
\end{tikzpicture}
\end{document}

This reveals that the left path is a loop, which explains the faint dots on the left. It is a loop because you run through the curved part from top left to bottom right and then through the horizontal part.

This also shows that the horizontal curve is not at all necessary here, for the fills, you could just do

\documentclass[10pt,multi=False,border=5pt,tikz,class=scrartcl]{standalone}
\usepackage{xcolor}

\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\usetikzlibrary{arrows.meta}
\usetikzlibrary{patterns}
\usepgfplotslibrary{fillbetween}
\begin{document}
\begin{tikzpicture}
  \begin{axis}[grid=both, axis equal=false, ymin=-1, ymax=6, xmin=-30, xmax=30,
    minor tick num=1,axis lines = middle,
    label style={font=\small,at={(ticklabel cs:1.1)}},
    tick label style={font=\footnotesize}]
    \addplot [thick,dashed,samples=50, domain=-30:0,name path=p1] {sqrt(abs(x))};
    \addplot [thick,samples=50, domain=0:30,name path=p2] {sqrt(abs(x))};
    \addplot [thick,dashed,name path=p3] coordinates {(-25,5) (25,5)};
    \path [name path=left, %thick,draw=blue,-latex,
    intersection segments={of=p1 and p3,sequence=L2}];
    \path [name path=right, %thick,draw=red,-latex,
    intersection segments={of=p2 and p3,sequence=L1}];
    \addplot [pattern=north east lines,pattern color=blue,opacity=.8] fill between [
        of=left and right,reverse=false];
    \draw [fill=gray,opacity=.5] (0,3.5) ellipse [x radius=12.25, y radius=.1];
    \addplot+ [only marks,mark=*] coordinates { (-25,5) (25,5)}
        node [pos=0,above right] {\footnotesize\textcolor{blue}{$B=(-25,5)$}}
        node [pos=1,above left] {\footnotesize\textcolor{blue}{$A=(25,5)$}};
    \node [right] at (11,3) {$r=x=y^2$};
  \end{axis}
\end{tikzpicture}
\end{document}

to get the first output above.