Excluding some special cases, computing the critical inverse-temperature $β_c$ of a mixed $p$-spin spin glass model is a difficult task. The only known method to calculate its value for a general model requires the full power of the Parisi formula. On the other hand, an easy application of the second moment method to the partition function yields an explicit lower bound $β_m\leq β_c$ to the critical inverse-temperature. Interestingly, in the important case of the Sherrington-Kirkpatrick model $β_m=β_c$. In this work we consider the multi-species spherical mixed $p$-spin models without external field, and characterize by a simple condition the models for which the second moment method works in the whole replica symmetric phase, namely, models such that $β_m=β_c$. In particular, for those models we obtain the value of $β_c$.