The space filling curve you are looking for does not exist.

Assume by contradiction that such a space filling curve $\gamma:I\rightarrow [0,1]^2$ exists. Since $\gamma$ intersects each curve $f_t\subset [0,1]^2$ only once, the preimage $\gamma^{-1}(f_t)$ is either a point or an interval. The curve $\gamma$ being space-filling, that preimae can't be a point. It is therefore an interval and, in particular, of positive measure.

Letting $t$ vary, we have constructed an uncountable family of disjoint subsets of $[0,1]$, all of whom have positive measure: contradiction!

The space filling curve you are looking for does not exist.

Assume by contradiction that such a space filling curve $\gamma:I\rightarrow [0,1]^2$ exists. Since $\gamma$ intersects each curve $f_t\subset [0,1]^2$ only once, the preimage $\gamma^{-1}(f_t)$ is either a point or an interval. The curve $\gamma$ being space-filling, that preimage can't be a point. It is therefore an interval and, in particular, of positive measure.

Letting $t$ vary, we have constructed an uncountable family of disjoint subsets of $[0,1]$, all of whom have positive measure: contradiction!