= Solution
The coordinate change has inverse $x=\tilde x-\tilde y^2+\varepsilon$, $y=\tilde y$. Thus the coefficient function needed after changing variables is
$$
\boxed{\tilde f_\varepsilon(\tilde x,\tilde y)=f(\tilde x-\tilde y^2+\varepsilon,\tilde y).}
$$
The PDF's displayed formula has an undefined subscript $f_\alpha$ and writes the forward change in the argument. It is not the coefficient pullback by the stated coordinates; the inverse expression above is the one used in transforming the equation. The forward <polynomial> composition, if intended instead, has the same type of uniform bounds, but is not that pullback.
For $|(\tilde x,\tilde y)|\le1/2$ and $0\le\varepsilon\le1/32$, the inverse image lies in the fixed compact subset specified by $|x|\le25/32$, $|y|\le1/2$, whose Euclidean radius is less than one. A <real analytic function> extends holomorphically to a complex neighborhood of this compact set: local extensions agree on overlaps by uniqueness, and a finite covering gives a common smaller neighborhood. The <polynomial> inverse maps a fixed small complex neighborhood of the closed half-ball into that neighborhood, uniformly in $\varepsilon$. The extensions are bounded there by a common constant $A$. Applying the multivariable <Cauchy estimate> on equal-radius polydiscs gives
$$
\boxed{\|\partial^\beta\tilde f_\varepsilon\|_\infty\le A\beta!R^{-|\beta|}=\gamma\beta!\zeta^{|\beta|}.}
$$
This is <uniform analyticity under polynomial coordinate changes>. The argument gives a compact margin and a common complex radius, not merely separate analyticity for each parameter.
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