I never heard of starlings that were not birds. But here Dickens uses it to mean a pier of a bridge.

"There were states of the tide when, having been down the river, I could not get back through the eddy-chafed arches and starlings of old London Bridge; then, I left my boat at a wharf near the Custom House, to be brought up afterwards to the Temple stairs. I was not averse to doing this, as it served to make me and my boat a commoner incident among the water-side people there. From this slight occasion, sprang two meetings that I have now to tell of."
From a site about London Bridge:
In 1666, the houses on London Bridge were saved from the Great Fire of London, thanks to an earlier fire in 1633 - which had destroyed the houses near to the north bank, so there was nothing for the fire to get hold of. In the mid 1700s, the houses on the bridge were removed completely, and a larger middle arch was created, by removing one of the piers (or starlings).