= Electrostatic calculation for parallel currents
Put wire 1 on the $z$ axis and wire 2 at $x=d$, with positive $x$ pointing away from wire 1. Let each wire have stationary positive <linear charge density> $+\lambda$ and moving <Electron> line density $-\lambda$ in the lab. For equal currents, both <Electron> populations have drift <velocity> $+v\widehat{\mathbf z}$, so the signed conventional current is $I=-\lambda v$.
First consider wire 2's positive charges in their own rest frame, which is the lab. Wire 1 has zero total line charge there, so its external electrostatic field is zero. The positive-charge <force> per length is therefore $\boxed{f_+=0}$.
Next use the <Electron> rest frame $S'$ moving at $+v\widehat{\mathbf z}$. Wire 1's ion line density becomes $\lambda_+'=\gamma\lambda$. Its <Electron> density becomes $\lambda_-'=-\lambda/\gamma$: the <Electron> rest density is lower than its lab density. The transformed total is
$$
\lambda_1'=\gamma\lambda-\frac\lambda\gamma
=\gamma\lambda\frac{v^2}{c^2}.
$$
Equivalently this follows from the <four-current> transformation $\lambda_1'=\gamma(\lambda_{1,\rm net}-vI_1/c^2)$, with $\lambda_{1,\rm net}=0$ and $I_1=-\lambda v$. By the electrostatic field of a <line charge>,
$$
\mathbf E_1'=\frac{\lambda_1'}{2\pi\epsilon_0d}\widehat{\mathbf x}.
$$
A wire-2 <Electron> is at rest in $S'$ and feels only the electric <force>, $F'_{e,x}=-e\lambda_1'/(2\pi\epsilon_0d)$, toward wire 1.
Apply the supplied transverse <relativistic force transformation> with the <Electron>'s lab <velocity> $\mathbf u=\mathbf v$. Its denominator is $\gamma(1-v^2/c^2)=1/\gamma$, so $F'_{e,x}=\gamma F_{e,x}$, or $F_{e,x}=F'_{e,x}/\gamma$. There are $\lambda/e$ <Electrons> per unit lab length. Thus
$$
\boxed{f_-\equiv\frac{F_{-,x}}\ell
=\frac\lambda e\frac{-e\lambda_1'}{2\pi\epsilon_0d\gamma}
=-\frac{\lambda^2v^2}{2\pi\epsilon_0c^2d}
=-\frac{\mu_0I^2}{2\pi d}.}
$$
Adding $f_+=0$ gives exactly the attractive magnetic result. This <electrostatic-frame derivation of parallel-wire forces> uses electrostatics in each test carrier's own rest frame; no magnetic <force> acts on that stationary test carrier there. The other wire can still carry current in that frame. Transform the <force> per <Electron> first and then count <Electrons> per lab length: transforming a <force> per unit length as if it were a particle <force> would introduce an incorrect extra contraction factor.
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