This cache is at the intersection of the "new" US-24 and the "old" US-24, now called Business 24 and MO-154. I gather that Paris was bypassed around 1970 (based on the Missouri highway map archive, found here.) Right where the main highway curves to the northeast, there is still a short section of the old road that was just never taken up. I'm not old enough to remember what it was like to share such a narrow road with the high traffic volumes the main roads see, but I'm glad things are a little wider now.
It is not advisable to park on US-24 to find this cache. There is adequate shoulder width to safely park on Business 24, and the cache itself is far enough off the road that you will not be exposed to oncoming traffic.
If I told you what you were looking for, it would spoil the cache. It is sufficient to say that there is only room for a log, so bring your own writing implement.