Quite often, whenever there are problems during a download cycle -say, your browser developed a conflict during the download of a particular package, and if the error appeared to resolve itself to the browser, no flag (or notification to the user) would be raised. HOWEVER, if the issue ACTUALLY was not fixed completely, and only appeared to be fixed to the browser, then the likelihood of the app installing correctly is remote. The first sign of such a problem would be exactly what you experienced. "Problem parsing..." This message, by itself, will tell you nothing.
But the answer to your question, in THIS case (download issues), would certainly not do your device any harm. Before even posting the question, you should have done SOME secondary tests, to reveal additional information on the situation. The only way that the conditions you described could damage your device, would be if whatever you downloaded was actually OPENED, after download. The first thing I would check, if concerned about possible damage, would be the source of the error. This can be done by examining whatever WAS downloaded. A simple text editor could try and open whatever is there, if it is complete. If it isn't, it SHOULD be deleted right away.
Other items to check are any other things you downloaded from that source. Since it seems you left everything alone, you should be quite fine. Next time run at least 1 test before beginning to panic ;).