I love this question, as it’s something I’ve never thought about before, but it’s the kind of thing I would think about.
The relationship between distance from the Sun, a, and orbital period, P, is not linear. So there is a chance that at some distance the two properties are changed in proportion to each other in a way that arrives at the coincidence you ask about. If the relationship were linear, then the relationship between a and P, (and thus a “local parsec” and a “local light-year”) would always be the same.
If we stick to worrying about our own Sun, we can take advantage of the fact that the square of the orbital period squared is proportional to the cube of the distance from the Sun. In other words, P[sup]2[/sup] is proportional to a[sup]3[/sup], and so P is proportional to a[sup]1.5[/sup]. For example, Venus is 0.723 times as far from the sun as the earth, and the orbital period is 0.723[sup]1.5[/sup] * 365.24 days = 224.7 days.
To find out the distance necessary to get the “local light-year” to match the “local parsec”, you need to find the distance where the ratio of the two has diminished to 1/3.262 of what it is for Earth. (Because here it’s 3.262, and we want it to be 1). Since the “local light-year” is proportional to P, it is therefore proportional to a[sup]1.5[/sup]. The “local parsec” is simply proportional to a. So the equation of interest reduces to simply:
parsecs/light-year = a/P = a/a[sup]1.5[/sup] = 1/3.262
This reduces to a[sup]1/2[/sup] = 3.262, or a = 10.64 AU.
So a planet 10.64 times as far from the Sun as the Earth would have the coincidence you describe. There are no planets there, but Saturn is closest at 9.54 AU. Let’s run a quick check on Saturn’s orbital parameters and see if that ratio is indeed close to 1:
Saturn is 9.54 times as far from the sun as Earth, so a “Saturn parsec” would also be 9.54 times as long. which is 31.1 “Earth” light-years). Saturn’s orbital period is 29.46 Earth years, and so a “Saturn light-year” is 29.46 “Earth light-years”. Those are, as expected, pretty close to 1. A “Saturn parsec” is 1.06 “Saturn light-years”, compared to the 3.26 ratio here on earth.
So, if we orbited our Sun at 10.64 AU (just outside Saturn), we would have defined the parsec and the light-year to be the same distance. For a different star, you can’t use the Earth’s (or any other planter in our solar system) orbital period and distance as a basis to simple ratios, and would have to calculate them directly based on the mass of the star and the equations of motion.