Having a pair royal count for twelve is the cribbage equivalent of getting money for landing on Free Parking.
Concurring with everyone else who’s responded to the OP: you don’t score the triplets and the three component pairs; you score the triplets as six as a shorthand for scoring the three component pairs as two points each.
That’s what cribbage players do: with frequent combinations, like triplets, four of a kind, and double, triple, and quadruple runs, you just point out the total points for the whole shebang, rather than adding up the component parts that everyone is familiar with, that are the actual source of the points.
When you have A-2-2-3-3, you just say ‘quadruple run for 16’ or something like that, because nobody past beginner needs to have it explained that that’s four distinct runs of three plus two pair = 43 + 22 = 16. But those pieces are what you’re actually scoring; there’s no separate points for the whole thing.
So, since IT was incorrect, mom can put her blouse back on and Idle can remove his socks.
/official scoring
That’s what it’s called in French: in America, it’s called the Royal Sampler.
I’ve only heard that called a “double double run”.
It’s hard to tell from the picture, but I think a Royal Sampler is a busted flush that’s also three to the royal. At least that’s how the Stonecutters play it.
My nomenclature may be a bit nonstandard, since I’ve played mostly with family. But we score the same way everyone else* does, AFAICT, regardless of the words we use.
*OK, everyone else except Idle Thoughts and his friends. 
Yes, double-double is the standard nomenclature.
Then what do you call 2-2-3-4-K vs 2-2-3-4-5? The first is a double run of three; the second is a double run of four. Or at least that’s the way I was taught… I’ve never heard double-double.
That’s the only way I’ve ever seen it scored or played anywhere. (And I’ve played thousands of hands with different people, IRL, on online games, on my cribbage app on the phone, etc.) The OP’s variation is definitely way non-standard. The scoring is pretty standard, and about the only rule/scoring variation I can think of coming across is whether you play “muggins” or not. (ETA: Actually, the way match points are scored can vary for normal win vs skunk vs double skunk.) “Muggins” allows you to peg points off your opponent’s hand if they forget to count it. For example, if you miss the 15s in a run like 7-8-8-9 and only call it for 12 instead of 16, I can claim the four points by calling muggins. I personally don’t play this rule, as pretty much anyone past beginner level has the scoring figured out, and the only players I’d be using against are those still learning the game. Yes, I know it can come up in more advanced games through simple oversight (like missing the nobs or something), but I just don’t like the rule.
I would use the same terminology for those hands. A double-double run (of three) would be something like 6-7-7-8-8.
Agree with the majority, Mom is correct!
And my Mother taught me that 6-7-7-8-8 was a “double-double run”.
I remember calling a 6-7-7-8-9 as a “Long Double Run”
My mother always said the “Right Jack” probably because she thought “His Nobs” sounded perverted.
Yes, those are both double runs (one of three, the other of four.) Double-double is the, as the name implies, having two double runs in your hand, like 2-2-3-3-4. FWIW, the American Cribbage Congress uses the terminology like that, too (and that’s the terminology I’ve always heard. “Quadruple run” is a new one for me, but makes sense and is scored the same.)
“Pair royal” is definitely a term used, especially during pegging play–that’s at least how I normally hear it. You throw out a two; the pone (your opponent) throws out another two and pegs two for a pair. You throw out a two, and you peg six for “pair royal” (sometimes called a “prial.”) If the pone then throws out a two, they peg twelve for a “double pair royal.”
In scoring the hand and crib during the show, all that counts is:
nobs (1), pair (2), fifteens (2), run (1 point for each card in a run), flush (4 or 5; 4-point flushes are only possible in the hand, not the crib. All four hand cards must match suit. A crib flush must be the four cards in the crib plus the start/up card.)
There is no separate category for “pair royal” or “double pair royal” that is distinct from the simple pair category to get scored additionally. A pair royal is worth six because it contains 3 pairs in it, and a double pair royal is scored twelve because it has four pairs in it. Counting the pairs and the pair royal separately would be like counting a run of four as ten points, because it contains two runs of three within it.
ETA: Or, for that matter, a 5-card run as 22 points, because it contains, in addition to the 5-card run, two 4-card runs and three 3-card runs in it.)
I hope Mom doesn’t gloat too much 
It’d be like scoring 8 for a grand slam in baseball. 4 for the grand slam, and then one point for each of the runs.
Which brings up a question I’ve occasionally wondered: why isn’t a 4-card run scored as two overlapping 3-card runs (for 6 points rather than 4), and similarly, why isn’t a 5-card run scored as 3 overlapping 3-card runs (for 9 points rather than 5)?
Not that I expect anybody to change the rules - or that it would even be a good idea. But given the way everything else in cribbage is scored, this has always seemed anomalous to me.
Heck, if the run is 4-5-7-6-8 or 5-6-7-8-9, there’d be an extra four points for two fifteens and if they’re all the same suit, an extra five points for the flush - boom, 31.
A 29 point maximum is only a suggestion…
Sure! I didn’t want to complicate the matter by introducing the 15s. Just the 22 points for the 5 card straight. And you could have a lot of fun with a five-card flush! Five points for the five card flush and then four points for all the four-card flushes within it for a total of 25 points plus any additional 15s or knobs.
Muggins! It has six pairs.
(Yes, you had the points right so I don’t actually get any.)