I am wondering about two things:

1) Why was the distance at which 1AU would span a parallax of one second of arc picked as a unit of distance, and

2) Are there other units using minutes and degrees? (To create the terms Parmin and Pardeg obviously)

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Bailey’s second law; There is no relationship between the three virtues of intelligence, education, and wisdom.

The parsec was chosen as a plausible unit to measure distances in the Galaxy and beyond. It is a bit less than 1 and a half parsecs to the neares star, and the centre of the Galaxy is around ten thousand pc away, which makes for handy numbers. I think that the reason why the parsec and the light year, at a ca. 3:1 ratio, are competing to be the measure most widely used is that they are both extremely practical for the distances concerned.

By a happy coincidence, one million parsecs, or one megaparsec, is just about right to give intergalactic distances, Andromeda as the nearest major galaxy being about two thirds of a Mpc away. Same with million light years.

And no, I never heard of parmins and pardegs. If I calculate correctly, a parmin would be 512 trillion km or 0.08 light years or 3,300 AU - not very practical for measuring anything. Maybe for giving distances in the Oort cloud, which is reputed to span the space between 300 and 100,000 AU.

Ditto, the pardeg. My calculation gives it at 8.5 billion km or 57.3 AU - About 1 and a half times the semimajor axis of the orbit of Pluto. Too large to measure anything within the Solar system, way too short for measuring interstellar distances.

I think by defining units of anything, particularly distance, we tend to choose those which give "handy numbers" with the objects we investigate. It is not a coincidence how the yard was defined: human scale. Or the mile, or the km - they make for a good walk. Of course you could give the distance to Proxima Centauri as 3.99 x 10^13 km, but, you know, 4.24 light years or 1.3 parsec is just vastly more familiar and thus, practical.

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Non sunt multiplicanda entia praeter necessitatem.

As noted above there is no real need for new measures. The parsec came into being when we were still fairly new at using parallax to determine interstellar distances. But the term refers to our old system of measuring angles, using base sixty divisions. You could extend it (and maintain some historical accuracy) to describe parthirds, parfourths, and parfifths. A parthird would be a couple hundred lightyears (nearby stars). A parfourth would be about twelve thousand lightyears (nearby spiral arms). A Parfifth would be about three quarters of a million lightyears (good fraction of the distance to galaxies in the local cluster.

Please note that if you use these terms, NO ONE will know what you're talking about.

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Forming opinions as we speak

Parthird. I like that. It's 195.5 lightyears. If only Hans Solo had known about parthirds.

Oh no, not the Kessel run again...

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Non sunt multiplicanda entia praeter necessitatem.

Actually, parsecs were used to make the math for parralax "easier". By that it means no conversion necessary. Parallax calculation using one second of arc produces a distance of one parsec.

One parsec is exactly 3.26 light years.

The terms parsec and light years are used interchangably for distance (keeping in mind the size difference) on many scales.

One reason for it's prominence is, before the last century or so, the AU wasn't very well defined, and parrallax measurements are based upon it. By tying the unit of parrallax (the parsec) to an AU (via the parrallax calculations), the numbers would be unchanged even as the AU was better confirmed. So even if they were off by a factor of 2 for the AU, the distance to alpha centauri would still be about 1.3 parsecs. Only the light year conversion would be different.