Here is how I made those flower petals. This is the method I worked out. Someone else may know a better way.
First, wrap the thread around your hand, but instead of bringing it into the pinch as for a ring, bring the thread over the index finger and then into the pinch. This is the color that will show on the outer part of the petals. I used two shuttles, but perhaps this one could actually be a ball thread. You will hold this thread in place while weaving the other shuttle around it, so it feels a little like a cluny. Perhaps you could make a loom to hold the thread just like you do to avoid those cluny hand cramps.
Next, with the other shuttle, make an unflipped stitch around the left thread. (I had to make a few repeats before I could get a good photo.) If you are trying for one-sided/front-sided tatting appearance, make the halves of the stitch in reverse order.
Repeat back and forth, with the first stitches very close together, and then leaving more thread between sides as the V shape widens.
Once you have made enough repeats back and forth, end with a stitch on the left side. (Yes, that's an uneven number of stitches, but it works easier for me to start on the left.) Tension the stitiches so that they lie close together, and then begin making regular flipped stitches for the chain across the top of the petal.
Once you have enough stitches, tug this shuttle to tension the chain, then take the work off your hand and pull on the other shuttle to close the loop.
Then make one more stitch to bring the core thread down to the base of the petal. With the blue and gold threads, I had switched shuttles to make this stitch and with the pink and purple I did not. I'm not sure which way is best, though you would need to do it the second way if you were using a ball thread instead of two shuttles.