Data integration and cleaning have long been a key focus of the data management community. Recent research indicates the potential of large language models (LLMs) for such tasks. However, scaling and automating data wrangling with LLMs for real-world use cases poses additional challenges. Manual prompt engineering for example, is expensive and hard to operationalise, while full fine-tuning of LLMs incurs high compute and storage costs. Following up on previous work, we evaluate parameter-efficient fine-tuning (PEFT) methods for efficiently automating data wrangling with LLMs. We conduct a study of four popular PEFT methods on differently sized LLMs for ten benchmark tasks, where we find that PEFT methods achieve performance on-par with full fine-tuning, and that we can leverage small LLMs with negligible performance loss. However, even though such PEFT methods are parameter-efficient, they still incur high compute costs at training time and require labeled training data. We explore a zero-shot setting to further reduce deployment costs, and propose our vision for ZeroMatch, a novel approach to zero-shot entity matching. It is based on maintaining a large number of pretrained LLM variants from different domains and intelligently selecting an appropriate variant at inference time.