For Android 4.2.2, you generally need or slightly newer versions specifically tagged for older API levels (API 17).
Go to Settings > Security and check the box for Unknown Sources to allow installation of non-Store apps.
If you're looking to download the Google Play Services APK for any reason, you should be cautious and only do so from trusted sources. Here are some steps and considerations:
How to Download Verified Google Play Services APK for Android 4.2.2
You cannot download the "latest" version; it will crash on Android 4.2.2. You need a legacy version compatible with Android 4.2 (API Level 17).
(e.g., “My Files” or “Downloads”).