The rapid growth of user-generated social media content has created new opportunities for automated mental health monitoring. However, a significant resource disparity persists between high-resource languages such as English and low-resource languages such as Bangla. While English NLP benefits from large-scale pre-training corpora and mature transformer architectures, Bangla NLP faces challenges including limited labeled data, code-mixing, and dialectal variation. This study presents a comparative analysis of deep learning models (LSTM, BERT, and BanglaBERT) and parameter-efficient fine-tuning through Low-Rank Adaptation (LoRA), evaluated on over 18,000 social media posts. In addition to classification accuracy, we assess statistical reliability and computational efficiency. Experimental results show that English models substantially outperform Bangla models, with BERT-base achieving 98.51% accuracy and its LoRA-adapted variant achieving 98.19%, while reducing trainable parameters by approximately 98%. The robustness of these findings is confirmed through 5-fold stratified cross-validation and 95% confidence interval analysis (Mean: 98.77%, CI: 97.76%-99.59%). In contrast, BanglaBERT achieved 86.24% accuracy, reflecting a consistent 12% performance gap. These findings highlight the impact of linguistic resource availability on depression detection performance and emphasize the need for culturally grounded Bangla corpora and improved cross-lingual transfer strategies to promote more equitable AI-driven mental health systems.