“Parse Error : There is a problem parsing the package” while installing Android application

You need to save the APK to a public directory.

For example:

Android.OS.Environment.GetExternalStoragePublicDirectory(Android.OS.Environment.DirectoryDownloads).ToString();

References :
https://forums.xamarin.com/discussion/5272/stuck-on-running-downloaded-apk-there-was-a-problem-parsing-the-package
https://pupli.net/2016/06/19/install-application-programmatically-on-android/