Can I just shoot more photos with more overlap on each other without rotating at the lens nodal point? Or will the stitching software encounter problems properly "aligning" parallax affected photos together?
It depends on the scene you're trying to shoot, how you're trying to shoot it, and what kind of 360 panorama you're making.
Coverage and parallax are two separate things, so, no you can't "fix" parallax errors by shooting more coverage with more overlap. And yes, stitching programs can encounter alignment problems when parallax comes into play with nearby subjects.
Parallax error basically means that if you change the viewpoint between two shots, some objects will be placed differently with respect to other objects in the frame. A lightpole that was on the right of a car, might be to the left of it, if you shift the camera far enough. The closer the lightpole is to the camera, the larger the shift may seem. Because all the stitchers can do are warp images to get things to fit, this type of shift can't be accommodated for, and you'll get stitching errors. On the other hand, if you're just shooting landscape shots of the horizon and there's nothing nearby, there may be no discernible shift at all, and everything will stitch cleanly.
However, once you start using nearby subjects, or you begin shooting 360 panoramas indoors, parallax becomes much more critical and problematic. If you shoot without precisely rotating around the no-parallax point of the lens, you can end up with visible breaks in lines--stairwells, wall/ceiling seams, etc. won't "match up" correctly, because simple warping can't overcome the parallax issue of objects "jumping" relative to each other. At this point, you will need a panorama head.
In addition, if you are not just shooting 360 cylindrical panoramas, but 360x180 cubic/spherical panoramas, you may not only have to rotate around the NPP in yaw, but you may also have to rotate around it in pitch when shooting your zenith (straight up) and nadir (straight down) shots, and a tripod and panorama head are almost a necessity for that--unless you're a phenom at proprioception. Most of us aren't, and require a panorama head and tripod--or at least a plumbline--to do this kind of thing.
Footnote: I've gotten extremely lucky a time or two, while just freely handholding and shooting a 360x180 without any mechanical aids, but those times are rare, and the NPP of the lens (Sigma 8mm f/3.5 circular fisheye) was well-defined (gold ring), and the coverage was big enough that I only had to shoot four images rotated in yaw at 90-degree intervals.
Distance to the closest object
Let's say the closest object from your lens is a chair at 50cm. The different shots will be very different regarding this chair.
If the closest object is like 5m away the misalignment will be smaller.
So you have to decide or prepare the scene or your point of view to minimize this defect.
One option is to have this close object on the center of one shot so it is not shared on two shots.
This of course implies that you need to shoot this specific case with a wider angle lens so your chair is on one single shot at close range.
Lens physical length
Another variant you have is how close your nodal point is to the center of rotation of your tripod.
For example the Canon pancake lens has its nodal point closer to the center than a zoom lens.
Where is the rotation center of your tripod head? In some cases it can produce a very big difference.
If you need it or not... How about you go out and take your first panorama. One indoors, other outdoors and see if you need it?
So the answer is probably no, depends...