There are times when lighting originates from the cloud, from the ground, or when the bolt jumps from cloud to cloud. In all of these cases, it is more complex than that. Let's look at the most common case. The lightning starts with a dim, negatively charged spark called a stepped leader, moving downwards from the cloud towards the ground. As it nears the ground, it induces a strong positive charge on objects below. The positive charge is attracted to the negative charge of the stepped leader, and the charge moves upwards to the highest points. This is why tall objects are often struck by lightning, and why lightning rods work.
As the stepped leader gets closer, one or more dim, positively charged upwards leaders move up from the ground. When the upwards leader meets the stepped leader, the lightning channel if formed. Electrons flow from the negatively charged cloud to the ground, but it does not stop there. The charge will oscillate back and forth many times in a fraction of a second.
So to answer the question, the sparks start from both cloud and ground, meeting in the middle.