After knowing that you could easily connect multiple simple reactor/choke ballasts in parallel for adding more current to HID and fluorescent lamps, I am trying to see about looking for a safe method of connecting multiple autotransformer ballasts in parallel for running certain European specification HID lamps like European 700w 145v 5.4a mercury vapor lamps or European 1000w 145v 7.5a mercury vapor lamps properly on a 120v mains supply.

From what I understand, it seems safe to wire multiple identical autotransformer ballasts with the same OCV in parallel.

On the other hand, I have been told that when I wire different autotransformer ballasts in parallel like mixing HX, CWA, and CWI together, or even differently branded autotransformer ballasts of the same circuit type and different wattage autotransformer ballasts of the same circuit type, I end up getting phasing issues that can cause the ballasts involved to go up in smoke.

In that case, I just want to confirm that I can safely connect identical autotransformer ballasts in parallel and would appreciate some help with connecting different wattage autotransformer ballasts of the same circuit type (like CWA) in parallel.

If you wire the ballast outputs in parallel, their currents will sum up. But they won't sum up arithmetically, but it will be a vector sum. So the resulting current could be anywhere between the arithmetic sum to the arithmetic difference, depending on the phase shift between those contributors.
And all that with the load voltage resulting from the load and the phase of the resulting current.

Because information about the exact phase shift uses to be difficult to get, the only way to get somehow predictable result is to use ballasts of the same phase shift with at least similar OCV and similar load voltage rating. Then the total current almost matches tye arithmetic sum.
That means you the easiest to combine are inductor ballasts (the OCV is the input, both are inductors so phase is the same, otherwise you are combining just the inductances).
Close by are the HX, as they are of just inductive impedance as well, they have just one more variable - their OCV.

And difficult became the CWA or so. They differ not only by the OCV, but also by the amount of resonant voltage boost they are designed to use. And that affects the phasing very significantly (still all are lead, but the exact angle may vary 30 degrees in either direction). Plus there are various flux choke points in the core (designed to shape the overal load characteristic), which have their impact on the phase shift as well. So another one or even two variables to count with. So much, I would say it becomes impossible to combine anything else than ballasts of exactly the same model.