Full linking is hard for biblatex. So there is a good reason why only the year is linked by default. Have a look at https://tex.stackexchange.com/a/27615/35864 for a full-link solution that colours both parentheses for \textcite.
Alternatively, I could offer you this answer I once gave at golatex.de: https://golatex.de/biblatex-textcite-link-fehler-bei-einem-autor-t20366.html (at least I think it was me ...)
\documentclass{article}
\usepackage[british]{babel}
\usepackage{csquotes}
\usepackage[backend=biber,
style=authoryear-comp,]{biblatex}
\usepackage{hyperref}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@article{AuthorA2000,
author = {AuthorA},
title = {TitleA},
year = {2000}
}
@article{AuthorB2001,
author = {AuthorB},
title = {TitleB},
year = {2001}
}
@article{AuthorC2002a,
author = {AuthorC},
title = {TitleCa},
year = {2002}
}
@article{AuthorC2002b,
author = {AuthorC},
title = {TitleCb},
year = {2002}
}
@article{AuthorC2002c,
author = {AuthorC},
title = {TitleCc},
year = {2002}
}
@article{AuthorC2003,
author = {AuthorC},
title = {TitleC3},
year = {2003}
}
\end{filecontents}
\addbibresource{\jobname.bib}
\addbibresource{biblatex-examples.bib}
\makeatletter
\newrobustcmd{\blx@imc@transparentprinttext}[2][]{%
\ifblank{#2}
{}
{\ifblank{#1}
{\let\blx@theformat\@firstofone}
{\blx@getformat\blx@theformat{ffd}{#1}{}}%
\ifdefvoid\blx@theformat
{}
{\blx@theformat{#2}}}}
\blx@regimcs{\transparentprinttext}
\DeclareCiteCommand{\textcite}[\cbx@textcite@init\cbx@textcite]
{\gdef\cbx@savedkeys{}%
\citetrackerfalse%
\pagetrackerfalse%
\DeferNextCitekeyHook%
\usebibmacro{cite:init}%
\global\undef\cbx@lasthash
\global\undef\cbx@lastyear}
{\ifthenelse{\iffirstcitekey\AND\value{multicitetotal}>0}
{\protected@xappto\cbx@savedcites{()(\thefield{multipostnote})}%
\global\clearfield{multipostnote}}
{}%
\xappto\cbx@savedkeys{\thefield{entrykey},}%
\iffieldequals{namehash}{\cbx@lasthash}
{\iffieldequals{labelyear}{\cbx@lastyear}
{\csnumgdef{cbx@textcite@state@\the\c@citecount}{2}}
{\csnumgdef{cbx@textcite@state@\the\c@citecount}{1}%
\savefield{labelyear}{\cbx@lastyear}}}
{\csnumgdef{cbx@textcite@state@\the\c@citecount}{0}%
\stepcounter{textcitetotal}%
\savefield{namehash}{\cbx@lasthash}}}
{}
{\protected@xappto\cbx@savedcites{%
[\thefield{prenote}][\thefield{postnote}]{\cbx@savedkeys}}}
\DeclareFieldFormat{citehyperref}{%
\DeclareFieldAlias{bibhyperref}{noformat}% Avoid nested links
\bibhyperref{#1}}
\DeclareFieldFormat{textcitehyperref}{%
\DeclareFieldAlias{bibhyperref}{noformat}% Avoid nested links
\bibhyperref{%
#1%
\ifnumgreater{\value{citecount}+1}{\value{citetotal}}
{\ifbool{cbx:parens}
{\bibcloseparen\global\boolfalse{cbx:parens}}
{}}
{\ifnumequal{\csuse{cbx@textcite@state@\the\numexpr\c@citecount+1\relax}}{0}
{\ifbool{cbx:parens}
{\bibcloseparen\global\boolfalse{cbx:parens}}
{}}
{}}}}
\savebibmacro{cite}
\savebibmacro{textcite}
\renewbibmacro*{cite}{%
\printtext[citehyperref]{%
\restorebibmacro{cite}%
\usebibmacro{cite}}}
\renewbibmacro*{textcite}{%
\ifboolexpr{
( not test {\iffieldundef{prenote}} and
test {\ifnumequal{\value{citecount}}{1}} )
or
( not test {\iffieldundef{postnote}} and
test {\ifnumequal{\value{citecount}}{\value{citetotal}}} )
}
{\DeclareFieldAlias{textcitehyperref}{noformat}}
{}%
\transparentprinttext[textcitehyperref]{%
\restorebibmacro{textcite}%
\usebibmacro{textcite}}}
\makeatother
\begin{document}
First, single cites which are working: \textcite{AuthorA2000} and so on \textcite{AuthorB2001} and so on \textcite{AuthorC2002a} and so on \textcite{AuthorC2002b} and so on \textcite{AuthorC2002c}. This works!
\par
Now combined: Different Authors \\
Two: \textcite{AuthorA2000,AuthorB2001}\\
Three: \textcite{AuthorA2000,AuthorB2001,AuthorC2002a}\\
Works again! \par
Now combined, single author same year
The problem if linking is on: \textcite{AuthorC2002a,AuthorC2002b,AuthorC2002c} \par
Same Author, diff year: Same problem: \textcite{AuthorC2002a,AuthorC2003}
\parencite{knuth:ct:a,knuth:ct:b}
\printbibliography
\end{document}

Here is a solution that never links (colours) the brackets. For \cite/\parencite we can use Audrey's approach from hyperlink name with biblatex authoryear (biblatex 1.4b). But for \textcite only want to additionally link the name, so we patch the textcite bibmacro with xpatch to link the labelname.
\documentclass{article}
\usepackage[british]{babel}
\usepackage{csquotes}
\usepackage[backend=biber,
style=authoryear-comp,]{biblatex}
\usepackage{hyperref}
\DeclareFieldFormat{citehyperref}{%
\DeclareFieldAlias{bibhyperref}{noformat}% Avoid nested links
\bibhyperref{#1}}
\savebibmacro{cite}
\renewbibmacro*{cite}{%
\printtext[citehyperref]{%
\restorebibmacro{cite}%
\usebibmacro{cite}}}
\usepackage{xpatch}
\xpatchbibmacro{textcite}
{\printnames{labelname}}
{\printtext[bibhyperref]{\printnames{labelname}}}
{}{}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@article{AuthorA2000,
author = {AuthorA},
title = {TitleA},
year = {2000}
}
@article{AuthorB2001,
author = {AuthorB},
title = {TitleB},
year = {2001}
}
@article{AuthorC2002a,
author = {AuthorC},
title = {TitleCa},
year = {2002}
}
@article{AuthorC2002b,
author = {AuthorC},
title = {TitleCb},
year = {2002}
}
@article{AuthorC2002c,
author = {AuthorC},
title = {TitleCc},
year = {2002}
}
@article{AuthorC2003,
author = {AuthorC},
title = {TitleC3},
year = {2003}
}
\end{filecontents}
\addbibresource{\jobname.bib}
\addbibresource{biblatex-examples.bib}
\begin{document}
First, single cites which are working: \textcite{AuthorA2000} and so on \textcite{AuthorB2001} and so on \textcite{AuthorC2002a} and so on \textcite{AuthorC2002b} and so on \textcite{AuthorC2002c}. This works!
\par
Now combined: Different Authors \\
Two: \textcite{AuthorA2000,AuthorB2001}\\
Three: \textcite{AuthorA2000,AuthorB2001,AuthorC2002a}\\
Works again! \par
Now combined, single author same year
The problem if linking is on: \textcite{AuthorC2002a,AuthorC2002b,AuthorC2002c} \par
Same Author, diff year: Same problem: \textcite{AuthorC2002a,AuthorC2003}
\parencite{knuth:ct:a,knuth:ct:b}
\printbibliography
\end{document}

biblatexcitations is hard, take a look at https://tex.stackexchange.com/a/27615/35864 – moewe Jun 24 '18 at 09:34style=authoryeardestroys some of the features ofstyle=authoryear-compwhich you are using. – moewe Jun 24 '18 at 09:44