Will using synthetic oil in my 2013 Grand Caravan increase times between oil changes?

Either oil (synthetic vs conventional) can be used in any vehicle. Regardless of which you choose, the most important factors to the oil you select are this:

• What weight oil is the engine designed for. This can affect both oil pressure, and complete engine oiling.

• Is your engine a flat tappet, or roller tappet cam engine (very
  important). Flat tappet engines were left behind in the 90s, but if you have a flat tappet engine, a higher zinc content oil is required,
  especially on new engines, and high performance engines.

Neither of your vehicles are going to be flat tappet.

If you choose to run fully synthetic motor oil, generally speaking I'd argue that you can increase intervals as long as you monitor the oil level. I tend to increase my intervals by 20-40%. If you choose to increase intervals, you should also run a higher grade oil filter. This all depends on the engine, but modern engines like those described in your post are probably fine.

Using synthetic oil in and of itself would not increase change intervals. There are products on the market which would allow you to do this, though, such as Mobil 1 Annual Protection. According to the product literature, it's good for either 1 year or 20k miles. I believe Amsoil has something similar (or their suggestion is for running a longer change interval). (Note: I have no affiliation with these products.) Other than products like these, you should change the oil at the manufacturer's suggested change interval. Synthetic, while being superior to conventional oil in almost every way, is not a magic bullet.