$R_{e q}=R_1+R_2$

$2 R\left(1+\alpha_{e q} \Delta \theta\right)=R\left(1+\alpha_1 \Delta \theta\right)+R\left(1+\alpha_2 \Delta \theta\right)$

$2 R\left(1+\alpha_{\mathrm{eq}} \Delta \theta\right)=2 R+\left(\alpha_1+\alpha_2\right) R \Delta \theta$

$\alpha_{\mathrm{eq}}=\frac{\alpha_1+\alpha_2}{2}$

Parallel :

$\frac{1}{R_{e q}}=\frac{1}{R_1}+\frac{1}{R_2}$

$\frac{1}{\frac{R}{2}\left(1+\alpha_{e q} \Delta \theta\right)}=\frac{1}{R\left(1+\alpha_1 \Delta \theta\right)}+\frac{1}{R\left(1+\alpha_2 \Delta \theta\right)}$

$\frac{2}{1+\alpha_{\mathrm{eq}} \Delta \theta}=\frac{1}{1+\alpha_1 \Delta \theta}+\frac{1}{1+\alpha_2 \Delta \theta}$

$\frac{2}{1+\alpha_{\mathrm{eq}} \Delta \theta}=\frac{1+\alpha_2 \Delta \theta+1+\alpha_1 \Delta \theta}{\left(1+\alpha_1 \Delta \theta\right)\left(1+\alpha_2 \Delta \theta\right)}$

$2\left[\left(1+\alpha_1 \Delta \theta\right)\left(1+\alpha_2 \Delta \theta\right)\right]$

$=\left[2+\left(\alpha_1+\alpha_{22}\right) \Delta \theta\right]\left[1+\alpha_{\mathrm{eq}} \Delta \theta\right]$

$2\left[1+\alpha_1 \Delta \theta+\alpha_2 \Delta \theta+\alpha_1 \alpha_2 \Delta \theta\right]$

$=2+2 \alpha_{\mathrm{eq}} \Delta \theta+\left(\alpha_1+\alpha_{22}\right) \Delta \theta+\alpha_{\mathrm{eq}}\left(\alpha_1+\alpha_2\right) \Delta \theta^2$

Neglecting small terms

$2+2\left(\alpha_1+\alpha_2\right) \Delta \theta=2+2 \alpha_{\mathrm{eq}} \Delta \theta+\left(\alpha_1+\alpha_2\right) \Delta \theta$

$\left(\alpha_1+\alpha_2\right) \Delta \theta=2 \alpha_{\mathrm{eq}} \Delta \theta$

$\alpha_{\mathrm{eq}}=\frac{\alpha_1+\alpha_2}{2}$