report does not provide marks for the part level. You could use scrreprt and scrlayer-scrpage instead of report and fancyhdr:
\documentclass{scrreprt}
\usepackage[autooneside=false]{scrlayer-scrpage}
\pagestyle{headings}
\automark[chapter]{part}
\chead{}
\ihead{\leftmark}
\ohead{\rightmark}
\renewcommand*{\chapterpagestyle}{headings}% use headings page style also for chapter pages
\usepackage{mwe}
\begin{document}
\tableofcontents
\part{First Part}
\blinddocument
\part{Second Part}
\blinddocument
\end{document}

If scrreprt should by more lookalike to report you can use option emulatestandardclasses:
\documentclass[emulatestandardclasses,autooneside=false]{scrreprt}
\renewcommand*{\chaptermarkformat}{\thechapter\autodot\enskip}
\pagestyle{headings}
\automark[chapter]{part}
\chead{}
\ihead{\leftmark}
\ohead{\rightmark}
\renewcommand*{\chapterpagestyle}{headings}% use headings page style also for chapter pages
\usepackage{mwe}
\begin{document}
\tableofcontents
\part{First Part}
\blinddocument
\part{Second Part}
\blinddocument
\end{document}

If you want a head separation line, you can use option headsepline. See the KOMA-Script manual for more information on the configuration of KOMA-Script classes or package scrlayer-scrpage.
Note: You should not use titlesec together with a KOMA-Script class like scrreprt. You should use the KOMA-Script features to configure the sectioning commands instead.
To get the same with report you have to patch the class:
\documentclass{report}
\usepackage[autooneside=false]{scrlayer-scrpage}
\pagestyle{headings}
\automark[chapter]{part}
\chead{}
\ihead{\leftmark}
\ohead{\rightmark}
\usepackage{xpatch}
\makeatletter
\xpatchcmd{\@part}{\@endpart}{%
\partmark{#2}%
\@endpart
}{}{}
\makeatother
\renewcommand*{\partmarkformat}{\thepart.\enskip}
\renewcommand*{\chaptermarkformat}{\thechapter.\enskip}
\xpatchcmd{\chapter}{\thispagestyle{plain}}{}{}{}% dont use pagestyle plain for chapter pages
\usepackage{mwe}
\begin{document}
\tableofcontents
\part{First Part}
\blinddocument
\part{Second Part}
\blinddocument
\end{document}

Note: You cannot combine fancyhdr and scrlayer-scrpage. So if you use scrlayer-scrpage to configure the page headings like shown above, you must not load fancyhdr!
But if you absolutely want to use fancyhdr you can replace the scrlayer-scrpage code of the last example by some fancyhdr code, e.g.,
\documentclass{report}
\usepackage{xpatch}
\makeatletter
\xpatchcmd{\@part}{\@endpart}{%
\partmark{#2}%
\@endpart
}{}{}
\makeatother
\usepackage{fancyhdr}
\pagestyle{fancy}
\newcommand*{\partmark}[1]{\markboth{\thepart.\enskip \MakeUppercase{#1}}{}}
\renewcommand*{\chaptermark}[1]{\markright{\thechapter.\enskip \MakeUppercase{#1}}}
\renewcommand*{\sectionmark}[1]{}
\lhead{\leftmark}
\rhead{\rightmark}
\xpatchcmd{\chapter}{\thispagestyle{plain}}{}{}{}% dont use pagestyle plain for chapter pages
\usepackage{mwe}
\begin{document}
\tableofcontents
\part{First Part}
\blinddocument
\part{Second Part}
\blinddocument
\end{document}
Note: If you are using package titlesec you have to use \renewcommand to (re)define \partmark. And patching \@part may fail, if you use titlesec to redefine \part. And patching \chapter may fail, if you use titlesec to redefine \chapter.
\bfshould not be used anymore. Use either\bfseriesor\textbf{…}or\normalfont\bfseries. Andscrreprtwith optionemulatestandardclassesis still not exactly the same as the standard classreport. But you can change most things and use, e.g, packagegeometryto change the margins. For the line in the page header see my note aboutheadsepline. – Schweinebacke Oct 16 '17 at 10:07\bf, but it is encoutered in the ToC, apparently – Tristan Oct 16 '17 at 10:13\bfand should not. Make a bug report to the author of that package. Related: https://tex.stackexchange.com/questions/516/does-it-matter-if-i-use-textit-or-it-bfseries-or-bf-etc KOMA-Script itself does not use\bffor the ToC. However, if using a KOMA-Script class is a problem in your case, see thereportsolutions in my answer. – Schweinebacke Oct 16 '17 at 10:15\bfsomewhere after all. You are great. Last question: the header still appears at ToC and bibliography, how can I remove it on those pages? – Tristan Oct 16 '17 at 10:22\pagestyle{plain}and\thispagestyle{plain}. Note, that ToC etc. are chapters too and atscrreprttherefore use\thispagestyle{\chapterpagestyle}themselves. So you either have to redefine\chapterpagestyletoo or use something like\AfterTOCHead{\thispagestyle{plain}\pagestyle{plain}. See the KOMA-Script manual for more information about these commands. – Schweinebacke Oct 16 '17 at 10:29