Excessive numbers living in one single family home

To be honest with you, your best bet is to have the HOA hold the owner's feet to the floor. You can't really touch the renters but you can make the owner responsible for the renters actions. If there are rules and fines for each violation, then send a copy of the violations and fines to the owner.
You can't restrict the number of people living in a house or if it is used as rental property. Your HOA can control how many parking spaces are handed out if PRIVATE parking exists. The noise levels may be under the HOA control as well if in the Convenants and Restrictions or other adopted rules. We have a No noise after 10 pm, Contractors can't start work before 7 AM and must be gone by 6 PM, and no pool use after 10 PM.
Essentially, until you can get the owner's attention to the situation, your NOT going to be able to do much. Ultimately, it is the OWNER's responsibility to control what is going on in their own home.
I can relate to your side and the owner's side in this situation. I rented my Patio Home out after I moved out of my HOA. My tenants had motorcycles and later had a baby Emu living in the backyard!!! So you could imagine my HOA's reaction and mine! Unfornately, it did take me 6 months to kick my tenants out due to "Rental rights". So understand, the owner may not be aware of the situation, and once they are, may not be able to kick the tenants out in a timely enough manner. It can take up to a year in some states to remove bad renters! So you can hope the owner will take action if notified properly and professionally.

Myron, it is pretty obvious who the "renters" are. One approach to try (I stress the word "try") getting a solution is to write your congessmen and the "govenator". Another is to contact your city counsel and pass an ordinance to prevent owners from renting to persons who may not have proper documentation. Google Farmers Branch TX and see what they have done. IMO, its a step in the right direction albeit a difficult one.

Section 7.1. Covenants and Restrictions. The following covenants, restrictions, conditions and limitations as to the use and occupancy which shall run with the land shall be binding upon each Unit Owner, his heirs, tenants, licensees and assigns.

A. Purpose of Property. No unit shall be used for any purpose other than as a residence site as follows:

(1) For a single family, meaning a group of one or more persons each of whom is related to the other by blood, marriage or adoption who are living together and maintaining a common household, but excluding more than one married couple and excluding two or more parents (not married to each other and not themselves parent and child) who have their children or stepchildren living with them.

(2) In the alternative, as a residence site for persons who do not constitute such a “family” but subject to the following restrictions on the total number of persons (including children) occupying any unit: Any two bedroom unit may be used or occupied by no more than four persons; and any three bedroom unit by no more than six persons, until and unless the Association adopts rules or regulations altering this requirement and no such rule or regulation may be adopted except by the members of the Association at a meeting duly called for that purpose.

(3) In accordance with the frequent approach in zoning codes of protecting values in residence districts by prohibiting the use of single family residences for roomers and boarders, and in order to provide similar protection for the owners of units., it is hereby provided that no roomers or boarders shall be permitted.