I saw a thread about hotels with family / quad rooms, started a few years back, and it is still a useful resource. Often recommended.
If someone needs other arrangements than the usual double, twin or single, the choices become much less. Triples are as few as quads, although they're also often needed. A family/group of 5 would need a triple and a double - in Paris the maximum allowed occupancy in a hotel room is 4 people so there will be no room taking all 5 together. Kids are counted from age 2 - under age 2 you might get a baby cot. A family/group of 6 has more options; either 2 triples or a quad and a double. Apartments are useful for families and groups, but they are usually rented on a weekly basis, but at least 4-5 days.
Inspired by the thread about the quadruple rooms, I started listing hotels with triple rooms. Unfortunately I couldn't come up with such a comprehensive work and my list is much shorter (some districts are missing altogether), but hopefully it will be amended. I have looked at hotels which I know (maybe stayed at but at least visited) from the years when I worked in Paris, or from more recent trips.
Anyhow, here is the list, according to districts/arrondissement - hope it will also develop into a useful thread.
Hotel Select http://paris-hotel-select.com
Hotel Aramis http://aramis-paris-hotel.com/
Hotel Aviatic http://aviatic-hotelsaintgermain.com
Hotel du Quai Voltaire http://www.quaivoltaire.fr/
Hotel Concortel http://www.concortel-paris-hotel.com
Hotel Place de Clichy http://hotel-paris-moulin-rouge.com
Hotel de Paris http://parishoteldeparis.com/
Hotel de Maubeuge http://paris-hotel-maubeuge.com/
Hotel Saint Quentin http://paris-hotel-saint-quentin.com/
Hotel Marciano http://www.marcianohotel-garedunord.com/
Hotel de France http://hotel-defrance-paris.com/
Holiday Inn Paris Montparnasse http://parishotelparnasse.com/
Hotel du Petit Louvre http://hotel-paris-petitlouvre.com
Hotel Belfast http://www.hotelbelfastparis.com
Hotel Damremont http://damremont-paris-hotel.com/
Hotel Palma http://www.paris-hotel-palma.com