Recent synthetic speech detection models typically adapt a pre-trained SSL model via finetuning, which is computationally demanding. Parameter-Efficient Fine-Tuning (PEFT) offers an alternative. However, existing methods lack the specific inductive biases required to model the multi-scale temporal artifacts characteristic of spoofed audio. This paper introduces the Multi-Scale Convolutional Adapter (MultiConvAdapter), a parameter-efficient architecture designed to address this limitation. MultiConvAdapter integrates parallel convolutional modules within the SSL encoder, facilitating the simultaneous learning of discriminative features across multiple temporal resolutions, capturing both short-term artifacts and long-term distortions. With only $3.17$M trainable parameters ($1\%$ of the SSL backbone), MultiConvAdapter substantially reduces the computational burden of adaptation. Evaluations on five public datasets, demonstrate that MultiConvAdapter achieves superior performance compared to full fine-tuning and established PEFT methods.