Too long for a comment. Actually I do not think that it is written anywhere but these kind of counterexamples are usually provided by the Ornstein-Uhlenbeck semigroup, generated by $Delta+Bx cdot nabla$, where $B$ is a matrix. Assuming that all eigenvalues of $B$ have negative real parts, then an invariant measure $mu$ exists (and is given by a Gaussian density). It turns out that the angle of analiticity in $L^p$ of the invariant measure can be computed exactly and can be smaller than $pi/2$, even for $p=2$. This can be found in a paper by Chill, Fasangova, Pallara and myself.
Chill, R.; Fašangová, E.; Metafune, G.; Pallara, D., The sector of analyticity of the Ornstein-Uhlenbeck semigroup on (L^p) spaces with respect to invariant measure, J. Lond. Math. Soc., II. Ser. 71, No. 3, 703-722 (2005). ZBL1123.35030.
To obtain similar examples in unweighted spaces one has only (but patiently) to compensate the weight thus obtaining an operator with a linear drift and a quadratic potential. This works however only from dimension $2$ on; in the one dimensional case the operator is always self-adjont but still the angle of analitycity is different from $pi/2$ in $L^p$, for $p$ different from $2$. There is also a paper by E. Priola and myself dealing with non-analytic Markov semigroups where one finds other examples.
Metafune, G.; Priola, E., Some classes of non-analytic Markov semigroups, J. Math. Anal. Appl. 294, No. 2, 596-613 (2004). ZBL1067.47055.
It would be nice to have a direct approach, avoiding the detour.
There are examples but you need singularities or unbounded coefficients; the uniformly parabolic case with regular coefficients is indeed a perturbation of the laplacian. In 1D, if you perturb the harmonic oscillator $D^2-x^2$ by a linear drift $bxD$, the angle of analyticity depends on $b$.