Hi!
I've never heard the phrase "a party of one," but I found the following link which may help you:
Urban Dictionary: party of one
But it really depends on context because if you're making a restaurant reservation, "a party of one" would mean that it will only be you sitting at the table. "A party of two" would, likewise, mean that it would be you and one other person sitting at the table.
I'm not sure if there are any other meanings for "a party of two".
Hope this helps!