Automated Knowledge Graph Completion (KGC) is a crucial task in the semantic web community, focused on discovering missing structured information within Knowledge Graphs (KGs), such as identifying links between entities or classifying relation types. The continually improving performance of pre-trained Large Language Models (LLMs), with their inherent ability to learn world knowledge, has shown significant promise for KGC. However, as KGs evolve with new factual knowledge, continual fine-tuning of such models becomes necessary, making them vulnerable to catastrophic forgetting and incurring significant computational and storage overhead. In this paper, we propose a dynamic Parameter-Efficient Fine-Tuning method that introduces a novel and effective combination of masking and growing strategies for continual KGC, addressing a core limitation of existing methods: static architectures that suffer from parameter saturation over time. Our method enables LLMs to continually adapt to evolving KGs while preserving previously acquired knowledge. It supports knowledge transfer, mitigates catastrophic forgetting, and incrementally expands model capacity as needed. Evaluation is conducted in two continual learning settings, task-incremental learning for link prediction and class-incremental learning for relation extraction. Experimental results show that the proposed method outperforms both rehearsal-based and rehearsal-free baselines, offering an effective and scalable solution for continual KG modeling.