We give a combinatorial proof of the following theorem: The number of partitions of $N$ into distinct parts in which odd-indexed (even-indexed) parts are even is equal to the number of partitions of $N$ into parts congruent to 2,3,7 (mod 8) (1,5,6 (mod 8)). This provides a new combinatorial view of the infinite products appearing in the little G\"ollnitz partition identities. We also show that a finite version of the theorem involves lecture hall partitions in which in which odd-indexed (even-indexed) parts are even, giving an analog of the Lecture Hall Theorem of Bousquet-M{\'e}lou and Eriksson. This is joint work with Sylvie Corteel and Andrew Sills.