In order to do maintenance or patching on an Acquia site, you must be a technical contact for the site. In addition, you'll need to do the following:

• Set up access to the Drupal repository in order to check changes into the repository. For more information, see https://docs.acquia.com/cloud/manage/code/repository.
• Set up SSH access to connect directly to the server to look at error logs, etc. All connections to Acquia servers must be encrypted with SSH. For more information, see https://docs.acquia.com/cloud/ssh/enable.

What kind of access is needed to make changes to the parent of the multi-site install? Is different access needed to make changes to child sites of that parent?

On the administrative side, a multi-site install at Acquia is the same as a singe-site install. Any user who is a Technical Contact on the Acquia site containing a multi-site install has the ability to anything from the back-end on any of the sub-sites .

Access to the administrative paths on the subsites can be granted to others, but it cannot be taken away from Technical Contact on the multi-site at Acquia because those people will always be technically able to use Drush to generate one-time logins for other users on any of the multi-site instances.