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The main problem in Parliament Square is that it is still a one-way system. This promotes unsafe turning manoeuvres, the main risk to cyclists. A turning manoeuvre may not have been the cause of Emily Thornberry's particular crash, but the main work to be done is to return the square to two-way operation, with the principal junction of north-south and west-to-north-and-south movements (and vice versa in each case) located on the south-east corner, with the western and northern arms traffic-calmed (this could take multiple forms).
Hope she is ok. The Embankment Super Highway disappears and gives way to the cars o/s Parliament. From Lambeth Bridge to the square there's no cycle lane. Likewise from Westminster Bridge.
TFL haven't repainted the ASL by the pedestrian lights outside Parliament since road works 3-4 years ago!