A computation with magma reveals that this curve is isomorphic, as a genus two curve over the function field $\mathbb{C}(p_1,p_2,t_1,t_2,a)$, to the hyperelliptic curve given by the Weierstrass equation $$y^2=f(x):=4t_2^6x^6 + 4p_1t_1^2t_2^4x^5 + (p_1^2t_1^4t_2^2 - 4p_1t_1^2t_2^4a - 4p_2t_1^2t_2^4)x^4 + (-2p_1^2t_1^4t_2^2a - 4p_1p_2t_1^4t_2^2 + 4p_1t_1^4t_2^3)x^3 + (2p_1^2t_1^6t_2 + p_1^2t_1^4t_2^2a^2 - 4p_1^2t_1^2t_2^4)x^2 - 2p_1^2t_1^6t_2ax + p_1^2t_1^8 $$ This defines an Hyperelliptic curve for any value of the paramenters such that the discriminant of the equation is non-zero.

The discriminant is zero when $t_1=0$ or $t_2=0$ or $p_1=0$ (some cases one should consider apart) or a very large of huge degree irreducible polynomial in the parameters is zero.

A computation with magma reveals that this curve is isomorphic, as a genus two curve over the function field $\mathbb{C}(p_1,p_2,t_1,t_2,a)$, to the hyperelliptic curve given by the Weierstrass equation $$y^2=f(x):=4t_2^6x^6 + 4p_1t_1^2t_2^4x^5 + (p_1^2t_1^4t_2^2 - 4p_1t_1^2t_2^4a - 4p_2t_1^2t_2^4)x^4 + (-2p_1^2t_1^4t_2^2a - 4p_1p_2t_1^4t_2^2 + 4p_1t_1^4t_2^3)x^3 + (2p_1^2t_1^6t_2 + p_1^2t_1^4t_2^2a^2 - 4p_1^2t_1^2t_2^4)x^2 - 2p_1^2t_1^6t_2ax + p_1^2t_1^8 $$ This defines an hyperelliptic curve for any value of the parameters such that the discriminant of the polynomial $f(x)$ is non-zero.

The discriminant is zero when $t_1=0$ or $t_2=0$ or $p_1=0$ (some cases one should consider apart) or a very large of huge degree irreducible polynomial in the parameters is zero.