I wouldn't until the attachment is relatively established with the new LSA - it could be construed as sabotaging the new support or the child could undermine the new person with 'but biscuitus doesn't do it this way'. It might just have been that because you were such an integral part of their lives as their child's 1:1. And if you are no longer that close, the overtures of friendship might not occur.
That said, if you have common ground other than their child, once a new lsa is in place I would think there is no reason why you couldn't have a friendship provided boundaries were in place regarding the child's professional support. Kind of like when teachers end up teaching their friends children. It happens so it cannot be a complete taboo.
I'm an LSA at an infant school. A child I have supported one-to-one for three years is moving on to junior school. His parents made overtures of friendship to me last year, but accepted my professional 'distance'. We have a lovely working relationship.
I'm not sure how to handle the situation, should they try again to become friends once he has left the school and is no longer in my care. I like the parents and the child, and would happily be friends with them, but would it be wise? I'm thinking both in terms of for me, and for the child - he will need to make a new attachment to a new LSA.