Using hiking sticks or not is a personal preference, and in this case would also be dependent on the conditions. I am generally a one stick person but found two useful with the icy conditions in January. We saw many people without sticks - some struggling, some not - but I would recommend taking at least one, particularly if you are accustomed to using it. Cannot comment on afternoon parking other than when we were leaving the trail in the early afternoon, the parking lot was full. It is my understanding that the earlier you go, the better.

If you coming from Phoenix you will "feel" the traffic on your way to this point when hitting 89A as there is a main attraction on the way before the W Fork (Sliding Rock Park). It will give you a pulse of what is going on ahead. Might not be a bad idea to postpone for later your visit to the W. Fork if you are not doing the whole thing, why? Well, be aware that the three first miles of the total 8 miles trail (even less) has the most iconic places to see and great spots to rest under the vertical rocky wall with water running at your feet (it's on the first encounter with the river with the trail going left (but you need to take a right following the current. It's like 200 meters beautiful walk under a ceiling of rocks. You will be surprise how many people miss this cue. After that you can go back and follow the trail back). With that in mind and due the short version of the walk, you don't need to be there early and better late when the people have mostly left (later than noon of course). Now, By coming from Phoenix (South going North) on 89A, you can pass the parking lot of the trail and find (200-300 meters ahead) a side spaces used for parking (on your side of the road). It's unofficial but not illegal to park there (check for the signs that prohibit to do so close to the parking lot. Its a narrow side of the road space for maybe 15-20 cars the most (used as a last resource as it is hard to turn back to the official parking lot). Be aware that the lines of cars waiting to park are in line oriented North to South in case you want to wait in line. Now, If you are coming from Flagstaff, the entrance to the official parking lot will be on your side and also the line of car waiting to enter. But obviously, by coming this way you will have the chance to see the open spaces for parking on the other side of the road 200-300 meters before the parking lot, so you can turn around on the entrance to the parking lot (there is room for it) and go for those beside parking spots. I had the Red Rock Pass which doesn't works for official parking lots (you need to pay anyway for those parking's). There is no signs saying which Pass you need to park on the side of the road where is not signs prohibiting to do it and I saw no tickets on any of the cars that park there, including me. As long you are not in the actual road in this unofficial parking area you should be ok.