Doing this is possible, but a number of conditions must be met:
1) The Sub-Account needs to have access to the API. This means it must have a valid API Password (Set in subscription->Change Password in the ASM UI), and it must have API Credits
2) The Sub-Account needs to have access to the Parent accounts monitors. Otherwise it will be severely limited in what it can do in the API. This can be set in the parent account by
a. Login as admin to create subaccount
b. Login as that subaccount to activate the account
c. In the admin account go to "Share Access" -> "Share access to log and graphs" and add the email address of the subaccount and choose which folders they are allowed access to. From here you can limit the Sub-Account to a particular set of folders, or to all folders
3) When submitting the API requests (e.g. rule_get), the field "acct" must be filled in with the parent account's email address. Otherwise, the parent account's monitor's will not be accessed by the API request.