Double sections and dominating maps

As is well-known, given the complex sphere P^1 minus two points, there exist nonconstant holomorphic maps from the plane into this set, the simplest example of which is given by applying the exponential map and then composing with a M"obius transformation taking 0 and 1 to the two given punctures. Likewise, given the sphere minus one point, we can map the plane into this set by simply applying directly a M"obius transformation taking 1 to this puncture. In this paper we prove a parametrized version of this result.