Background
Reciprocal Gamma function as an entire function
The reciprocal Gamma function, \begin{equation} \begin{split} \frac{1}{\Gamma(z)} &= e^{\gamma z} \prod_{n = 1}^{\infty} \left( 1 + \frac{z}{n} \right) e^{z/n}, \\ \gamma & = \lim_{n \to \infty} \left[ 1 + \frac{1}{2} + \dotsc + \frac{1}{n} - \log n \right], \end{split} \label{eq:entire} \tag{1} \end{equation} is entire, and of order 1 and maximal (infinite) type by Stirling's formula (e.g., B. Ja. Levin's text [Lev64], Chap. 1, Sec. 11, p. 27). In particular, we write $\displaystyle \frac{1}{\Gamma(-n)} = 0$, $n \in \mathbb{N}_0 = \mathbb{N} \cup \lbrace 0 \rbrace$.
Parabolic Cylinder Functions
The Weber differential equation can be written in either of the forms
\begin{align} - \frac{d^2 y}{d^2 z} + \left( \frac{z^2}{4} - \frac{1}{2} \right) y(z) &= \nu y(z), \quad z \in \mathbb{C}, \quad \nu \in \mathbb{C}, \label{eq:ourWeber} \tag{2.i}\\ - \frac{d^2 y}{d^2 z} + \left( \frac{1}{4} z^2 + a \right) y(z) &= 0, \quad z \in \mathbb{C}, \quad a \in \mathbb{C}. \label{eq:theirWeber} \tag{2.ii} \end{align} These forms are equivalent under the rule \begin{equation} a = - \nu - \frac{1}{2} \label{eq:anuequiv} \tag{3} \end{equation} and \eqref{eq:anuequiv} is assumed throughout. For my purposes, writing the parameter as $\nu$ is best, and this convention is also used in the reference of W. Magnus, F. Oberhettinger, and R. P. Soni [MOS66]; we admit, however, that the $a$-notation for the parameter is commonly used (e.g., P. Dean's paper [Dea66], F.W.J. Olver's text [Olv97], and N. M. Temme's contribution to the Digital Library of Mathematical Functions [Tem19]).
The parabolic cylinder functions are solutions to the Weber differential equation. A particular solution, denoted $D_{\nu}(z)$ in $\nu$-notation and $U(a, z)$ in $a$-notation, is (for present purposes) the solution of the Weber differential equation with the initial conditions (e.g., [MOS66], Sect. 8.1., p. 324, and [Tem19], Sect. 12.2(ii)), \begin{align} D_{\nu}(0) &= \frac{2^{\nu/2} \sqrt{\pi}}{\Gamma \left( - \, \frac{\nu}{2} + \frac{1}{2} \right)}, & \frac{\partial}{\partial z} (D_{\nu}(z)) \big\vert_{z = 0} & = - \, \frac{2^{(\nu + 1)/2} \sqrt{\pi}}{\Gamma \left( - \, \frac{\nu}{2} \right)} \tag{4.i} \label{eq:inits}\\ U(a, 0) & = \frac{\sqrt{\pi}}{2^{a/2 + 1/4} \Gamma \left( \frac{3}{4} + \frac{1}{2} a \right)}, & \frac{\partial}{\partial z} (U(a, z)) \big\vert_{z = 0} & = -\, \frac{\sqrt{\pi}}{2^{a/2 - 1/4} \Gamma \left( \frac{1}{4} + \frac{1}{2} a \right)} \tag{4.ii} \end{align} (This solution is asymptotic to $z^{\nu} e^{-z^2/4} = z^{-a - (1/2)} e^{-z^2/4}$ as $z \to +\infty$, see [Olv97], Chap. 6, Sec. 6, p. 207.) In the sequel, we will abuse terminology and call $D_{\nu}(z) = U(a, z)$ "the" parabolic cylinder function.
By standard continuation-of-parameter results (e.g., [Olv97], Chap. 5, Sect. 3, Thm. 3.2, p. 146]), for any fixed $z$, the parabolic cylinder function is holomorphic in the parameter, since the initial conditions are holomorphic, and the coefficients of the system are holomorphic in each variable. Since $\nu$ is unrestricted, the parabolic cylinder function is entire in the parameter.
The Question
I am interested in the order and type of the parabolic cylinder function in the parameter, for any fixed $z$. For example, for $z = 0$, the initial condition \eqref{eq:inits} is essentially the reciprocal Gamma function, so $D_{\nu}(0)$ (or $U(a, 0)$) is an order-1, maximal-type function in the parameter.
I have cobbled together a proof that for any fixed $z$ in $\mathbb{C}$, the function $D_{\nu}(z)$ (or $U(a, z)$) is order-1 and maximal-type in the parameter, but the proof methods are distinctly standard. Therefore, with overwhelming likelihood, someone has discovered it before.
Question: What references discuss the order and type of the parabolic cylinder function in the parameter?
Motivation:
I (and my coauthor) need the fact that for any fixed $z$, the parabolic cylinder function has infinitely many zeros in the parameter. (OK, we only really need it for positive $z$, in which case the result goes back at least to [Dea66], but to make the paper self-contained, we wished to give a proof in the paper.)
Knowledge of the order and type is one way to prove this fact: if we supposed, by way of contradiction, that $D_{\nu}(z)$ was an entire function of finite order in $\nu$ with only finitely many zeros, it must be of the form
\begin{equation}
P(\nu) \exp (Q(\nu)), \quad P(\nu), Q(\nu) \text{ polynomials.}
\end{equation}
If both order and type are known, we are done immediately: a (nonconstant) function of this form must be of mean type (i.e., finite nonzero type), hence not of maximal type.
Even if only (an upper bound on) the order is known, we can get a contradiction by using the known decay of the parabolic cylinder function as $\nu \to - \infty$ ($a \to + \infty$), e.g. [MOS66], Sec. 8.1.6, p. 332: if $\displaystyle |\arg(- \nu)| \leq \frac{\pi}{2}$, for bounded $|z|$,
\begin{equation}
D_{\nu}(z) \approx \frac{1}{\sqrt{2}} \exp \left[ \frac{\nu}{2} \log(-\nu) - \frac{\nu}{2} - z \sqrt{-\nu} \right] \left\lbrace 1 + O (|\nu|^{-1/2}) \right\rbrace.
\end{equation}
(The authors of [MOS66] use $\sim$ for asymptotic expansions with no error terms and $\approx$ for asymptotic expansions with error terms, so I believe that "$\approx$" can be read as "$=$" in the case, but I am not 100% sure.)
References:
[Dea66] P. Dean. “The constrained quantum mechanical harmonic oscillator”. In: Mathematical Proceedings of the Cambridge Philosophical Society 62 (Apr. 2, 1966), pp. 277–286. Link to Cambridge U. Press
[Lev64] B. Ja. Levin. Distribution of zeros of entire functions. Vol. 5. Translations of Mathematical Monographs. Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman. American Mathematical Society, Providence, R.I., 1964.
[MOS66] Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni. Formulas and theorems for the special functions of mathematical physics. Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52. Springer-Verlag New York, Inc., New York, 1966.
[Olv97] F. W. J. Olver. Asymptotics and special functions. Reprint of the 1974 original [Academic Press, New York]. A K Peters, Natick, MA, 1997.
[Tem19] N. M. Temme. "Chapter 12 Parabolic Cylinder Functions." Digital Library of Mathematical Functions. Version 1.0.22; March 15, 2019. Link to DLMF