Indeed, we can see below that there is a limitation at the frequency of R/L in the differentiator section of the PID bode plot, and if the RPAR value was infinite we wouldn’t see it.

That’s not the point at all. Gmin works strangely.
Here is an example where I define the actual part - a parallel resistor. I find the conductivity, then the real (actual) conductivity and by finding the inverse I determine which resistor is in parallel with the inductance. My opinion is that something is wrong with Gmin.
I suggest you use .opt Gmin=0 for now.

.subckt pid_comp in out
XU1 in prop Gain A={km}
XU2 in mid1 Gain A={1/tau_i}
XU3 mid1 integ Integ initcond=0
XU4 diff prop integ out Adder3
XU5 in mid2 Gain A={tau_d}
XU6 mid2 diff Diff
.ends pid_comp

.subckt Diff in out
G1 0 int in 0 1
L1 int 0 1 Rpar=1e15
E1 out 0 int 0 1
.ends Diff

.subckt Integ in out
.param initcond=0
G1 0 int in 0 1
C1 int 0 1 ic=initcond
R1 int 0 1e18
E1 out 0 int 0 1
.ends Integ