Converting cascaded biquads to parallel biquads
You need to do partial fraction expansion.
Most of the time, the result will be 3 parallel sections with the same poles as the original filter but only with a real zero in each parallel section. The first section is just a single real scale factor.
$$H(z) = r0_0 + \frac{r0_1 + r1_1 \cdot z^{-1}}{a0_1 + a1_1 \cdot z^{-1} + a2_1 \cdot z^{-2}}  + \frac{r0_2 + r1_2 \cdot z^{-1}}{a0_2 + a1_2 \cdot z^{-1} + a2_2 \cdot z^{-2}} $$
Note that the &quot;a&quot; coefficients are the same as in the cascaded form, i.e. you only need to calculate the &quot;r&quot; coefficients.
This is all relatively straight forward if your poles are unique and complex conjugated pairs. Repeated poles and single real poles require special treatment.
The Matlab function residuez() can typically help with this although it can have numerical problems if the poles are too close to the unit circle. If you need to do it by hand take a look at https://ccrma.stanford.edu/~jos/filters/Partial_Fraction_Expansion.html
Since you only have two sections, the following approach should work

Write both versions (parallel and serial) as a single fraction
Discard the denominators, they are the same
Both numerators are a 4th order polynomial in $z^{-1}$, i.e. they have 5 coefficients.
Each power of $z^{-1}$ needs to have the same coefficient in both numerators.
That gives you five linear equations with 5 unknowns that you can solve for $r0_0, r0_1, r1_1, r0_2, r1_2$