Just a clarification first: Parsing is different from geocoding. If all you want is to get lat and long values filled in then regular google geocoding works without parsing.
If you really want parsing, then you might be able to do it with hook_civicrm_post and when $objectName == 'Address' you can parse and adjust the address there.
Note that just for ...
This should be possible out of the box. CiviCRM has mapping capabilities documented here https://docs.civicrm.org/user/en/latest/initial-set-up/mapping/
There is an extension using open source v google mapping if you want: https://github.com/eileenmcnaughton/org.wikimedia.geocoder
Once you have your mapping working, then you should be able to do a Search ...
It sounds like an incomplete install. You might need to run composer install from inside the ext/org.wikimedia.geocode folder.
But you don't need the extension if you just want to use basic geocoding in civicrm, you just need to set it up at administer - system settings - mapping and geocoding. The extension just adds more features.
UPDATE: It seems like the ...
For Wordpress, you can create a regular or smart group and use CiviCRM Directory to display those records in a directory.
It's basic alpha stage plugin so it depends on how much functionality you need. It will display all the core fields correctly. For custom fields it has an issue with not being able to retrieve them all from Civi in the edit screen.