Perinatal depression (PND) refers to a complex mental health condition that can occur during pregnancy (prenatal period) or in the first year after childbirth (postnatal period). Prediction of PND holds considerable importance due to its significant role in safeguarding the mental health and overall well-being of both mothers and their infants. Unfortunately, PND is difficult to diagnose at an early stage and thus may elevate the risk of suicide during pregnancy. In addition, it contributes to the development of postnatal depressive disorders. Despite the gravity of the problem, the resources for developing and training AI models in this area remain limited. To this end, in this work, we have locally curated a novel dataset named PERI DEP using the Patient Health Questionnaire (PHQ-9), Edinburgh Postnatal Depression Scale (EPDS), and socio-demographic questionnaires. The dataset consists of 14,008 records of women who participated in the hospitals of Lahore and Gujranwala regions. We have used SMOTE and GAN oversampling for data augmentation on the training set to solve the class imbalance problem. Furthermore, we propose a novel deep-learning framework combining the recurrent neural networks (RNN) and long short-term memory (LSTM) architectures. The results indicate that our hybrid RNN-LSTM model with SMOTE augmentation achieves a higher accuracy of 95% with an F1 score of 96%. Our study reveals the prevalence rate of PND among women in Pakistan (73.1%) indicating the need to prioritize the prevention and intervention strategies to overcome this public health challenge.