Nice! I like the parallaxing background. Regarding the jumping on start, do you have jump attached to "spacebar is currently down" or "spacebar is pressed"? The first will trigger every frame that space is down and the second will only trigger the first frame that it is down. My first guess at seeing this is that you used the first option which may be causing the problem?
The other thing that I can think of is that your dog already exists but hidden when the black screen is up and is interactible when you hit space. Are the black screen and gameplay different rooms or the same room?