We show that for any pair of long enough parallel geodesics in a random group G(m, d) with m generators at density d < 1/6, there is a van Kampen diagram having only one layer of faces. Using this result, we give an upper bound, depending only on d, of the number of pairwise parallel geodesics in G(m, d) when d < 1/6. As an application, we show that the minimal stable length of a random group at density d < 1/6 is exactly 1.