Since the page style elements are set when the page numbers are fully known, you can easily condition on whether or not you're on an odd/even page with \ifodd\value{page} <odd page>\else <even page>\fi:
\documentclass{article}
\usepackage{graphicx}
\usepackage{fancyhdr}
\pagestyle{fancy}% All pages will have a fancy page style
\fancyhf{}% Clear header/footer
\renewcommand{\headrulewidth}{0pt}% Remove header rule
\fancyhead[C]{%
\ifodd\value{page}
\smash{\includegraphics[height=.8\baselineskip]{example-image-a}}% Center of Odd page
\else
\smash{\includegraphics[height=.8\baselineskip]{example-image-b}}% Center of Even page
\fi
}
\fancyfoot[C]{\thepage}% Centre of footer
\fancypagestyle{firstpage}{%
% Page style for first page
\fancyhf{}% Clear header/footer
\renewcommand{\headrulewidth}{0pt}%
\fancyhead[C]{First page}% Header
\fancyfoot[C]{\thepage}% Footer
}
\usepackage{lipsum}
\begin{document}
\thispagestyle{firstpage}
\lipsum[1-50]
\end{document}
A separate firstpage page style is set up an used via \thispagestyle{firstpage}.
I've \smashed the images, but you could consider changing the \headheight to suit the size of your image inclusion.
twosidedocument. Seefancyhdrdocumentation how it handle page headers at such documents (section *10 An example of two-sided printing, page 11 -- 12). – Zarko Dec 16 '21 at 06:22