Place potatoes in a large saucepan and cover with cold water. Bring to the boil over medium-high heat, then reduce heat to medium and simmer for 12 minutes or until just tender. Drain, then transfer to a large bowl. Set aside for 15 minutes to cool.
Meanwhile, place basil, garlic, almonds, parmesan, oil, lemon juice and 1/4 cup water in a food processor and process until smooth.
Add pesto to potato, tossing to coat. Season, then divide potato and cabbage among plates. Scatter over extra almonds, basil and parmesan to serve.