I'm very flattered; but this is definitely an Ecalle/Baker level result that you just have to dig for 
Alexandre Eremenko used to always tell me to read more Baker--as he answered most of my questions with odd results from Baker, lol.
Thank you for putting me on this problem though. I've always wanted to use Mellin transform/Borel/Euler/Fourier stuff on tetration, and this problem has definitely helped approach this problem for the parabolic case--which I always thought would be untenable. And since it's so easy in the geometric case, I always knew there was a way to do it for the parabolic case--Especially because \(\eta \uparrow \uparrow z\) has a beautiful representation with Mellin transforms:
\[
\begin{align}
\vartheta(x) &= \sum_{k=0}^\infty \left(\eta^{\eta^{\cdots k+1\,\text{times}\,^\eta}}\right) \frac{(-x)^k}{k!} = \eta - \eta^\eta x + \eta^{\eta^\eta}x^2/2! -\eta^{\eta^{\eta^\eta}}x^3/3!\,+\,...\\
\Gamma(1-z) \cdot \eta \uparrow \uparrow z &= \sum_{k=0}^\infty \left(\eta^{\eta^{\cdots k+1\,\text{times}\,^\eta}}\right)\frac{(-1)^k}{k!(k+1-z)} + \int_1^\infty \vartheta(-x)x^{-z}\,dx\\
\end{align}
\]
Which is valid, at least, for \(\Re(z) > 0\).
I think we've just opened up a bunch of new techniques that I'm familiar with; but now we talk about the Abel function, rather than the inverse Abel function--and I think most of the heavy lifting's already done!
Alexandre Eremenko used to always tell me to read more Baker--as he answered most of my questions with odd results from Baker, lol. Thank you for putting me on this problem though. I've always wanted to use Mellin transform/Borel/Euler/Fourier stuff on tetration, and this problem has definitely helped approach this problem for the parabolic case--which I always thought would be untenable. And since it's so easy in the geometric case, I always knew there was a way to do it for the parabolic case--Especially because \(\eta \uparrow \uparrow z\) has a beautiful representation with Mellin transforms:
\[
\begin{align}
\vartheta(x) &= \sum_{k=0}^\infty \left(\eta^{\eta^{\cdots k+1\,\text{times}\,^\eta}}\right) \frac{(-x)^k}{k!} = \eta - \eta^\eta x + \eta^{\eta^\eta}x^2/2! -\eta^{\eta^{\eta^\eta}}x^3/3!\,+\,...\\
\Gamma(1-z) \cdot \eta \uparrow \uparrow z &= \sum_{k=0}^\infty \left(\eta^{\eta^{\cdots k+1\,\text{times}\,^\eta}}\right)\frac{(-1)^k}{k!(k+1-z)} + \int_1^\infty \vartheta(-x)x^{-z}\,dx\\
\end{align}
\]
Which is valid, at least, for \(\Re(z) > 0\).
I think we've just opened up a bunch of new techniques that I'm familiar with; but now we talk about the Abel function, rather than the inverse Abel function--and I think most of the heavy lifting's already done!