We're close to straining the parable beyond reasonable limits, but I anticipated this. Recall that the student has no proof the paper ever existed.
So if we tweak the story a bit where the student knows his friend's propensities and accuses him of using the paper for kindling (rather than the friend confessing), we indeed have a debate over existence.
I also anticipated the issue of probabilities that was mentioned earlier. Given the presence of the dog and that it's a windy day, the friend can easily argue that using the paper as kindling is the least likely answer. The wind is a more "natural" explanation than a need to invoke a person to make the paper disappear.
You're still missing the point. Your analogy makes no sense. Lets walk through and look at why:
Friend walks up to a picnic table where the student is sitting looking a bit distraught.
Friend, "Hey what's up?"
Student, "I wrote a paper this morning and left it on this table and now its gone!"
Friend. "What paper? I don't see any evidence for it that you even wrote it."
Student, "No, I did. I wrote a specific paper for this assignment."
Friend, "Okay, well first you have to prove that paper exists."
Student, "Okay, yea sure, here is some in my backpack." (Note #1: this is something I haven't seen a theist do: provide some agreed upon qualities to look for in a god).
Friend, "Okay, I see that paper so we can both agree on some qualities of what we are looking for. Now, how could you propose that we find your specific paper now that we have agreed we have agreed that paper in general exists?"
Student, "Well, we could look under the table. We could look in the dogs stomach. We could figure out which way the wind is blowing and go looking in that direction." (Note#2: this is another thing that theists fail to do: provide agreed upon ways to look for God...and even if found, we don't know that we've found him because of the lack of definition from Note #1)
Friend, "Cool, I'll help you look." (Note #3: this is another reason the analogy fails. Both people end up looking for the lost paper. With regards to God, theists claim to have found something called God while atheists can't seem to find it)
Friend, "Oh, here I found it!"
Student, "Oh awesome. Thanks."
Here is the analogy tweaked to make the theistic case more apparent:
Friend walks up to picnic table where the student is sitting, looking content,
Student, "Hey, I wrote this paper and its sitting on this table."
Friend, "Where? I don't see it."
Student, "Its right here."
Friend, "I still don't see it. What does it look like."
Student, "Well, its more of a person."
Friend, "What? But I thought you said it was a paper."
Student, "Not,
a paper,
the Paper."
Friend, "There's no other papers?"
Student, "Nope, this is the only one."
Friend, "I still don't see any evidence of this paper that you're talking about."
Student, "Well, like I said, its a person."
Friend, "Is it a person or is it a paper?"
Student, "Both."
Friend, "Ok, I'm confused. Are we talking about a paper that you wrote down on physical paper that has some sort of physical manifestation in the world, or is it just a paper that you wrote in your head?"
Student, "No its real."
Friend, "You didn't really answer my question"
And so on.