I wanted to explore the idea of this family since I haven't found any art of Instructor Mooselini and Instructor Moosesha together yet, in spite of them being canon sisters; I find that absence a little weird, but then again, I seem to find that a lot of "logical" ideas for fan art are dropped in favor of, for instance, PaRappa and PJ engaging in black-belt romantic karate, if you catch my drift. Nevertheless, I think it's high time someone tries to take a stab at it. Using the Dazzeloids character guide as a base, I'll give each of their biographies with something of a Greenblat-esque flair.

inb4 "But female moose don't have antlers!" I'm going by Greenblat's design — I figured he figured that having them all with antlers is a better designation for their species artistically; I could rationalize the artistic license by having them be more like reindeer in PaRappa's universe. Perhaps they're all just pumped with testosterone!

In General

History: The Moose Family is a proud family of instructors, teachers, and mentors, otherwise passing knowledge and skills to those in need. Familiar with street-smarts as much as book-smarts, they feel it's important to learn any ways to survive in an unpredictable and often violent neighborhood. Nevertheless, outside of danger, they know when to take care of their needs and desires in grand style.
Good Side: The Moose Family shares knowledge almost like the internet, tracking down books, trimming out misinformation, and utilizing their knowledge to the best of their abilities, encouraging others to do the same.
Bad Side: The Moose Family generally has a bit of a mean-streak, many with a trigger temper quick to snap against any threats or stubborn fools. Though they have not used their antlers as weapons since the early modern period, a few are not above a bit of charging practice against a particularly obstinate foe.

Mooselini

(Algonquin: Moose, Italian: Mussolini)
History: In spite of her unfortunate namesake, Mooselini strives to be a kind-hearted yet firm teacher. Her career in driving instruction was once seen as a stepping stone in her path of instruction, but when she saw the reckless driving of many first-timers in PaRappa Town, she dedicated herself to keeping sanity on the road, even if it means having to spend days on end behind the wheel with minimal bathroom breaks.
Good Side: Though stern and often aggressive-looking, she's only out for her student's best, making sure that they keep on task and on-rhythm, especially in emergency situations.
Bad Side: So single-minded is she to the success of her students, that she might forget a few key rules of the road, such as closing the car doors.

Moosesha

(Algonquin: Moose, Hindi: Sesha)
History: The wild girl of the Moose family, Moosesha sought sport and adventure in her neighborhood, challenging anyone willing to street hockey. In her teens, after getting on the wrong side of an street fight, her mother sent her to the military to straighten out her attitude. Rising in the ranks, she quickly became a sergeant in General Potter's unit, and with a flair for loud command and finding the groove in anyone, she became the unit's drill sergeant, fulfilling her family's instructing destiny with flair and fashion in the mix.
Good Side: Having trained in the military, Moosesha is strong and steadfast, with an intolerance of failure and an encouragement of only the best in any private. She enjoys self-portrait photography in her down times.
Bad Side: Her wild attitude still resides within her, giving her a dismissive, casual attitude that sometimes conflicts with military regulations. Ultimately, she's a spitfire in or out of service, with her former calls to service showing daring plans of attack and sometimes reckless decisions.

Mama Mooselynn

(Algonquin: Moose, variant of Turkish: Selynn)
History: Mama Mooselynn was a police instructor before retiring from the force to raise a family. Mindful of the criminal rings that plague the otherwise peaceful PaRappa Town, Mooselynn tries to raise her children on the side of right, though this often means setting a super-strict example for the whole family, even her husband.
Good Side: Mooselynn is a natural at keeping criminals at bay, with a watchful eye of any suspicious activity, and a love of security devices and anti-criminal measures.
Bad Side: Such is her strictness that Mooselyn has a tendency to alienate people, potentially driving friends and acquaintances away should they do anything unruly in her eyes. She's also possessive of her daughters, all but preventing them from dating or even befriending strangers for fear of growing attached to those that could hurt them.

Papa Moosena

(Algonquin: Moose, Latin: Sena)
History: Papa Moosena is a mild-mannered elementary-school teacher specializing in history and social studies. He met Mooselynn in a shopping mart, both cautious to enter a relationship, but soon understanding each other's needs and marrying soon after. Though his friends regard him as hen-pecked, he feels it's only right that a woman like Mooselynn deserves the utmost care and respect for all she's done to keep this town from falling to pieces.
Good Side: Moosena is a natural in tending to children, even the annoying and gross ones. He also enjoys having children use his antlers as a jungle gym, even if his school employers forbid him from for fear of liabilities.
Bad Side: Without a mean bone in his body, Moosena tends to be a coward willing to give everything to protect his family, even his own life. Ultimately, he needs someone like Mooselynn to instruct him on how to keep his cool and toughen up.

Nana Moosenya

(Algonquin: Moose, variant of Greek: Senya)
History: Nana Moosenya was a government instructor specializing in training bank managers and other financial staff before retiring in her late-60s. Age has not been kind to Moosenya since then, as she starts to mistake ordinary people for potential trainees, as well as treat everything like banknotes and tender to file away, trade, invest in, and convert, even if it means bartering very strange items. Her daughter, Mooselynn, moved her to PaRappa town to tend to her in her senile years.
Good Side: She loves her family, treating even her son-in-law like a true son, though perhaps too dotingly. After a prolonged chain of deals to find the right ingredients, she can whip up mighty meals within minutes to feed an entire family of moose, and then some!
Bad Side: Such is her addled mental state that she will factor everything into a currency-like system sooner or later, which could lead to precious possessions traded away, and "banknotes" or items of supposedly equal value left in their place. She has facilitated such thefts before with shady characters, and so needs strict supervision to ensure family heirlooms remain and only inconsequential items bartered away.

Phew, I'm tired! Let's take a break.
Come on, please, for old time's sake?

There's also a PSD and transparent PNGs in zip available for download. Their usage is covered by Creative Commons Non-Commercial Share-Alike. Share any edits you do with these with the proper marks of what you did and why, and link back to me.
 

Instructors Mooselini and Moosesha's designs belongs to Rodney A. Greenblat/Interlink.
Original Moose characters and art belongs to CyberneticCupcake.