Face parsing is defined as the per-pixel labeling of images containing human faces. The labels are defined to identify key facial regions like eyes, lips, nose, hair, etc. In this work, we make use of the structural consistency of the human face to propose a lightweight face-parsing method using a Local Implicit Function network, FP-LIIF. We propose a simple architecture having a convolutional encoder and a pixel MLP decoder that uses 1/26th number of parameters compared to the state-of-the-art models and yet matches or outperforms state-of-the-art models on multiple datasets, like CelebAMask-HQ and LaPa. We do not use any pretraining, and compared to other works, our network can also generate segmentation at different resolutions without any changes in the input resolution. This work enables the use of facial segmentation on low-compute or low-bandwidth devices because of its higher FPS and smaller model size.