Parkinson's Disease (PD) is a progressive neurological disorder that significantly affects motor skills, typically altering a person's handwriting. This work investigates deep learning-based approaches for classifying Parkinson's Disease using images of handwritten spiral drawings. The study began with a transfer learning approach using EfficientNet, as well as a CNN-LSTM architecture that combines convolutional and recurrent layers for spatial- temporal feature modeling. However, both approaches individually yielded suboptimal results, each achieving only 50% classification accuracy. To overcome these limitations, a hybrid model is proposed that integrates MobileNet, a lightweight and efficient convolutional neural network, with a Long Short-Term Memory (LSTM) layer to capture both spatial and temporal dynamics in handwriting patterns. This MobileNet+LSTM architecture demonstrated significant performance gains, achieving 87% accuracy on the publicly available Parkinson's Drawings Dataset. These results suggest that combining transfer learning with temporal sequence modeling is a highly effective strategy for handwriting-based Parkinson's diagnosis, offering a non-invasive and scalable solution for early detection.