For $N$ spin-polarized fermions in a $1$-dimensional trap with strong attractive interactions an odd-even effect is found. The entanglement spectrum of the $1$-particle reduced density matrix of the system's ground state differs qualitatively for $N$ odd and $N$ even. Since this effect does not occur for bosons it is the result of the competition between the attractive interaction and the repulsive nature of Pauli's exclusion principle. In contrast to the spectrum, the fermionic $1$-particle ...