In one way you can just think of the fact that the cross product can be computed as a 3x3 determinant as a happy conincidence, or a handy way of remembering the formula. The formulas simply coincide.

Determinants for $n \times n$ matrices are defined inductively, independtly of the cross product. The cross product is defined differently in different textbooks, certainly depending on the level of the book. Some books define it by its formula and derive its geometric properties, while others define it by its geometric properties.

As you mention, there is a neat connection between the cross product and a $3 \times 3$ determinant through the area of a parallellepiped. It turns out that the area of a parallellepiped spanned by the vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ is $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$, and that it can be computed by taking the determinant of the matrix whose columns are $\mathbf{a}, \mathbf{b}, \mathbf{c}$. However, this does not really have anything to do with parallellepipeds. Once again, the formulas simply coincide. In general we have

$$\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \det[\mathbf{a} \, \mathbf{b} \, \mathbf{c}].$$

This can, rather boringly, be proven by simply varifying that both sides of the equations are the same.