That is the question when talking about pallet arbors. I was trained in the tradition that unless you are dealing with an extremely humid climate and a dust-proof case then the answer is no, you don’t oil pallet arbors. You really want the pallet arbor to be as free as possible to rock back and forth.
For those who oil pallet arbors I understand that an extremely light film is applied to the arbor as a rust preventative more than as a lubricant.
Whether you are a believer of oiling pallet arbors or not the following photo is NOT the way to do it.
You may want to click on the photo to get the full effect. What you see here is oil in the pallet arbor sink and flowing up on top of the pivot. Also you’ll notice oil on the top of the pallet fork. Whoever serviced this watch last used enough oil on the pallet arbor to take care of most of the gear train.