Yo R John:
I have an early '60's Type A that I have gone through and "restored," including having gone through the cleaning and relubrication process.
Before you start messing with the turntable, you might want to observe your father when he operates the unit. Although he is a little young for this to happen, he might be making a mistake in operation. There are 2 levers on a Type A that start it. In order for the arm to function automatically, the lever on the right, marked Automatic, must be used. It has 2 positions OFF, on left, and ON-REJECT on right. If he uses the left hand lever, marked Manual, and engages the position on right, marked ON, the unit will behave exactly as you describe. It will start turning, and do nothing else. You will have to pick up the arm and put it on the record. He could be doing this out of habit, from many years of having run it manually, or he could be making a mistake. Most constructors, (like me, even,) eschew automatic operation, and never consider the damned automatic function.
Normally, if the problem is grease or inadequate idler contact like Alec describes, the symptoms will be different. In the the automatic position the arm will lift and stop in the middle of its travel, and will hang up and be impossible to move or put down on record until you spin the turntable manually through the cycle. The turntable will slow down or stop during the cycle. I am not ruling this out, but if the wrong lever is being used and the unit is otherwiseOK, it may not be necessary to go through an overhaul.
I have some knowledge of these things, since I am charged with the care of my aged (93) mother. I do not mean to insult your father, but my mother sometimes has problems like this with her TV remote, or her small microwave, and the issue can sometimes be traced to a difficulty such as I have described, and taken care of with a reminder. You have to observe what happens. I just say to her, "Show me what happens, Mom," and she shows me, and then I can either fix the problem, sometimes by changing a remote function, or show her how to fix it. "If you want it to do that, just press the green button." If I do that, she's fine.
Like I said, I just think that an observation of the problem would be useful. Maybe you have already done so. But be sure that the right lever is being used when these symptoms occur. If it is a mechanical malfunction, we can go through some diagnostics to locate the trouble.
Good luck from another old constructor!