There are a couple ways to suppress labelname in citations. The hook
\AtEveryCitekey{\clearname{labelname}}
will delete the labelname list in every citation, but styles typically resort to printing fallback values. The authoryear style, for example prints the labeltitle field when labelname is unavailable. Suppressing all fallback values often results in spurious whitespace or punctuation.
One easy way around all this is to save labelyear into the shorthand field.
\documentclass{article}
\usepackage[style=authoryear]{biblatex}
\DeclareBibliographyCategory{citeyear}
\addtocategory{citeyear}{companion,knuth:ct:a}
\AtEveryCitekey{%
\ifcategory{citeyear}
{\edef\cbxlabel{\thefield{labelyear}}%
\restorefield{shorthand}{\cbxlabel}}
{}}
\addbibresource{biblatex-examples.bib}
\begin{document}
\textcite[10--11]{companion} showed that...
Filler \parencite[e.g.][10--15]{knuth:ct:a}.
Filler \textcite[10--11]{kant:ku}.
Filler \parencite[e.g.][10--15]{kant:kpv}.
\printbibliography
\end{document}

Note that the labelname still appears in \textcite. This is intentional. \textcite is analogous to \citeasnoun in the Harvard styles. Suppressing labelname in \textcite would be like omitting the noun in \citeasnoun. It makes no sense.
labelyear isn't a very meaningful citation label. So I would recommend using a shorthand based on the title or authors of the work. The above document cites two entries with shorthands from biblatex-examples.bib. Here's an excerpt from this file.
@inbook{kant:kpv,
shorthand = {KpV},
author = {Kant, Immanuel},
title = {Kritik der praktischen Vernunft},
shorttitle = {Kritik der praktischen Vernunft},
booktitle = {Kritik der praktischen Vernunft. Kritik der Urtheilskraft},
maintitle = {Kants Werke. Akademie Textausgabe},
volume = {5},
publisher = {Walter de Gruyter},
location = {Berlin},
date = {1968},
pages = {1--163},
}
@inbook{kant:ku,
shorthand = {KU},
author = {Kant, Immanuel},
title = {Kritik der Urtheilskraft},
booktitle = {Kritik der praktischen Vernunft. Kritik der Urtheilskraft},
maintitle = {Kants Werke. Akademie Textausgabe},
volume = {5},
publisher = {Walter de Gruyter},
location = {Berlin},
date = {1968},
pages = {165--485},
}
shorthandorlabelfields. This previous post provides some details. – Audrey Dec 10 '12 at 16:35shorthandis a field you add to thebibfile. It overrides the whole citation label. Referring to your two main papers only by year is a little ambiguous, but you can do this by clearing thelabelnamelist. Again, the answer depends on your style (authoryear?) and exactly what output you're wanting to achieve. You should add an example to your question to clarify all this. – Audrey Dec 10 '12 at 17:51