Are there known explicit parametrizations of smooth del Pezzo surfaces, say of degree $5$ or higher, as surfaces cut out by equations in some projective space?
The closest I've been able to find is on p. 42 of "Quantitative arithmetic of projective varieties" by Tim Browning (equation 2.21), where he describes a singular del Pezzo surface of degree $6$ by a system of $9$ quadrics in $\mathbb{P}^6$.
A smooth del Pezzo surface $X_d$ of degree $d\geq3$ has very ample anticanonical divisor $-K_X$. It is also the blowup $X\to \mathbb P^2$ in $k=9-d$ points $P_1,\ldots,P_k\in\mathbb{P}^2_{x,y,z}$ in general position, so we have that $-K_X=\pi^*\mathcal{O}_{\mathbb P^2}(3) - E_1 - \cdots - E_k$. Therefore you can write down the anticanonical embedding of $X$ by writing down a basis for the linear system of cubic polynomials in $x,y,z$ that vanish at $P_1,\ldots,P_k$. For example if $d=6$ then, up to a linear automorphism of $\mathbb P^2$, we can assume that the three points we blow up are $(1:0:0)$, $(0:1:0)$ and $(0:0:1)$ and the seven-dimensional linear system of cubic polynomials vanishing at these points has a basis given by $xyz,x^2y,xy^2,y^2z,yz^2,xz^2,x^2z$, which are the coordinates $b,a_1,\ldots,a_6$ in Will Sawin's comment above. (This also follows from the description of $X_6$ as a toric variety.)
For each $d$ this will give you an embedding $X_d\hookrightarrow \mathbb{P}^d$ of codimension $d-2$ where the $X_d$ is cut out by quadratic polynomials. (For $X_9=\mathbb P^2$ it gives the third Veronese embedding.) However there are many simpler embeddings that one can work with if you want to work with explicit equations. For example $X_6$ can also be described as a smooth hypersurface of bidegree $(1,1,1)$ in $\mathbb P^1\times \mathbb P^1\times \mathbb P^1$.
