I have had great success just using a parting tool. After turning the piece to size for the main portion of the blank body with the tenon I then make sure I know the length of the tenon and start my cut right there. Early on I would mark the line in pencil.
Go easy on your depth at first and measure often with your calipers.
For furniture joints, I use a parting tool to a diameter perhaps 1/16" larger in diameter, waste away with a spindle roughing gouge, push cuts with shallow gouge, or peeling cuts with a skew, then close in on the final diameter with slicing cuts with a skew.
Push it straight across as a scraper to start the cut; just a quick start to get it going. That initial scrape will prevent the parting tool from feathering the inside edge. Pull the parting tool back, tip it up, and bevel cut the remainder of the tenon. (Don't scrape the entire tenon, or you will put a lot of wear and heat on your parting tool.) Gently push the bevel forward and down as the tenon is cut. You will see and feel the action as you make the cut. Any non-woodturner who watches you will think you're a pro.
There is a certain raw pleasure in making tenons on a lathe with a parting tool. It is fun to watch the wood go down as you cut the tenon. Fun, as in two-year-old knocking down a tower of blocks fun. Try not to jump up and down and squeal with delight in front of others as you make tenons.
You didn't say why you want to make tenons. If it is for pen turning, then you're done. Measure carefully. It is easy to cut off, much harder to cut on.
Some people make dovetail tenons for mounting bowl blanks in a chuck for hollowing. Dovetail tenons have a slight angle to them to match the angles in the chuck jaw, making a better grip. Nova makes a special dovetail turning tool with just the correct angle to match their chucks (and probably most other brands of chuck). I do not have one. Instead, I use a diamond carbide tool to shape the angle after cutting the initial tenon. It doesn't have to be the exact perfect angle as long as the chuck has firm grip on the bowl. I like the diamond tool because it is more versatile. It has many uses, such as making decorative "grip" lines on pens.
Parting tool or bedan - depending on the desired length of the tenon.
Take small steps, measure frequently with calipers. If the end objective is a tenon that must fit into a mortise, test frequently, and gradually reduce the diameter to the target. I often make pieces with 1/2 or 5/8" tenons that fit into drilled mortises, so I drill a test hole in a scrap using the same bit that will be used to create the actual mortise to use as a gauge.
Sometimes, if the tenon is to be longer (say greater than 1/2"), give it a gentle taper, and do a test fit with the mortise that it must fit into. As the tenon approaches the desired final diameter, it will slip partway into the mortise, leaving a burnished line at the point on the taper that has the exact diameter - that makes it easier to turn the rest of the tenon to the desired diameter. The tip of the tenon will be slightly undersized - that's not a problem, and in fact putting a small chamfer on the end of the tenon helps with final assembly.
Related subject - glueing a tenon into a mortise. Ideally, the tenon will be just a smidge undersized to allow room for glue. Use a scrap of wood (I sometimes use plastic coffee stirs) to smear glue inside the mortise rather than on the tenon - that way, surplus glue is forced into the mortise. Glue applied to the tenon tends to be forced out of the joint. But a problem is that if the mortise is blind (ie, just a hole drilled into timber), air can be trapped in the mortise when you try to insert the tenon, and forcing the tenon into the mortise also squeezes glue out of the joint, potentially leaving the joint weak. One solution to this is to drill a weep hole in the side or bottom of the mortise using a very small bit (less than 1/16") to provide a path for air to escape when you push the tenon in. That hole will fill itself with surplus glue, so it doesn't require any repair later. But another approach is to use a dremel to cut a groove along the length of the tenon to provide an escape path for air. Again, the groove will fill with glue and won't actually weaken the joint.