Why ParallelTable warns where Table runs quietly?

I have a code involving ParametricNDSolveValue and FindRoot to find a trajectory [actually, trajectory of a line that everywhere is normal to magnetic field] passing through a given point {rn,zn} on the rz plane. First I call for ParametricNDSolveValue to find trajectory zs[z0] that starts from the point {0,z0} on the axis z:

dzdr$  [r_, z_] /; r < 1/10^5 = (  r (-9 Sin[2 z] + 4 Sin[4 z]))/(-9 + 9 Cos[2 z] - 2 Cos[4 z]);  dzdr$  [r_, z_] /;    r >= 1/10^5 = (2 r (9 BesselI[1, 2 r] Sin[2 z] -        2 BesselI[1, 4 r] Sin[4 z]))/(18 r + 2 r^3 -      9 r BesselI[0, 2 r] Cos[2 z] - 9 BesselI[1, 2 r] Cos[2 z] -      9 r BesselI[2, 2 r] Cos[2 z] + 2 r BesselI[0, 4 r] Cos[4 z] +      BesselI[1, 4 r] Cos[4 z] + 2 r BesselI[2, 4 r] Cos[4 z]);  pfun = ParametricNDSolveValue[{D[z[r], r] == dzdr$  [r, z[r]],     z[0] == z0}, z, {r, 0, 1.5}, {{z0, -(\[Pi]/2), \[Pi]/2}}]  zs[{r_?NumericQ, z0_?NumericQ}] := pfun[z0][r] 

At the second step I define function getZ0[{rn,zn}] that calls for FindRoot to find starting point z0 for the trajectory that passes through a given point {rn,zn}:

getZ0[{rn_?NumericQ, zn_?NumericQ}, z0Start_?NumericQ] := Module[   {sol}   , sol =     FindRoot[zs[{rn, z0}] - zn, {z0, z0Start, -(\[Pi]/2), \[Pi]/2}];   sol[[1, 2]]   ] getZ0[{rn_?NumericQ, zn_?NumericQ}] := getZ0[{rn, zn}, zn] 

Finally, I want to evaluate getZ0 on a rectangular grid using Table:

Table[{{rn, zn}, getZ0[{rn, zn}]}, {zn, -(\[Pi]/2), \[Pi]/2,    1/2 \[Pi]/2}, {rn, 0, 1, 0.5}] 

This works fine. However substuting Table with ParallelTable results in sequence of Warnings of the types: FindRoot::lstol, ParametricNDSolveValue::ndsz, InterpolatingFunction::dmval. However both routines seemed to give same results.

To say truth, the differential equation which is solved by ParametricNDSolveValue is singular in two poins (where magnetic field is zero). But I wonder why there are no warnings when I use Table rather than ParallelTable?