For very high-precision applications such as finishing, milling by CNC machines, jerk-bounded trajectories (that is, trajectories comprising polynomials of degree 3 of higher) are often used. If you search on Google Scholar using the term "jerk bounded", you can find loads of methods to plan such trajectories.
High-order polynomial trajectories (or splines) are not always necessary, however. For stiff articulated manipulators (such as position-controlled robots), they are usually stiff enough that using second-order (parabolic) polynomial trajectories is perfectly fine. The effect of (theoretically) unbounded-jerk is often negligible.
In your case, you mentioned that the errors got high when stopping and speeding up. I suspect that this might not come from the trajectory generation method you used.
If the robot you are using is custom-built, you might want to
Otherwise, you may check if this issue is related in any way to singularities.