I know many cruisers who have sailed long distances in Catalinas. They have a good reputation as a somewhat affordably-priced, middle-of-the-road production boat. It is a very good company, with a long history of building decent looking and coastal cruising sailboats. And though there have been many who have taken their Catalinas to far off places around the world, personally, I would not feel safe at sea in a major storm in even their largest model. They are just not built for that level of wind and sea conditions. Their rigging is undersized and their construction assumes they will be coastal cruising, maybe in rough seas, but not 20-30 foot seas for days on end, hundreds or even thousands of miles from help. That can quickly take a toll on any boat, let alone one not built for those conditions.
For sailing to the Caribbean from any Florida port, as long as you know what you are doing (as it sounds like your parents do), have the boat properly prepared to go to sea, pick the right time of year, have a complete understanding of the projected weather ahead, while you closely monitor any changes in forecast, 99 times out of 100, the Catalina will get you there safely. The Gulf Stream can be rough when the wind kicks up, but you should know the forecasted direction of the wind before heading off into it. When leaving FL, if you have a strong Norther, you don't cross the Stream. Wait until something more southerly fills in, or there is no wind. Turn on the motor and power through it while it is docile. After the Stream crossing, island hopping down to the Turks and Caicos is like coastal cruising, with the exception of your everyday tropical squalls, which during the day when you can see them, can be sailed around most of the time. At night, without radar, a little harder, but I will assume your parents have radar. Most boats making passages these days are equipped with radar.
From the T&Cs, they will have a lot of hard sailing to windward, which can make for some lumpy conditions when the wind gets up over 25 knots. Could be uncomfortable for a few days, but if they are going to the DR first, it will break it up. Most boats go straight from T&Cs to PR and then onto the VIs. Since the trip is all to windward, it can easily take 3 full days in a 36 footer to reach Puerto Rico. It's probably about 350-400 miles to San Juan from T&Cs, as the crow flies, but going to windward will add many additional miles. The tradewinds can get/be a little stronger if you try to make the trip after mid-December. So, I would recommend a November crossing, after hurricane season settles down and before the Christmas winds kick up in the eastern Caribbean. From San Juan, it is just a hop, skip and a jump to St. Thomas, albeit all to windward, and then into the BVIs for some of the most beautiful anchorages in the Caribbean. Oh man, why did I ever leave?