I am most grateful to Ray and everyone for trying to solve this long-standing problem. The photographs certainly suggest that Palmer's car was a standard(ish) Lowline "T53", not the "Slimline" design of the "T58".
As to which T53 he had, I think that will be very difficult to ascertain. Parnell's team acquired three production T53s for 1961 but almost immediately were racing four of them, and it looks to me that they may have made up a fifth at some point. Parnell later did the same with Lotus 21s and Lotus 24s, so the extra cars should not be a surprise. I have used the identity "P2" for the Palmer car. To reiterate my justification for this:
The name 'P2' is given here to the fourth standard T53 used by Yeoman-Credit in 1961. As well as the three cars that appear in Cooper records (F1-1-61, F1-2-61 and F1-4-61), they produced a streamliner known as "VR", one extra standard car and later a T56-based car. When the T56-based car appeared at Monza, Denis Jenkinson noted that this brought their team strength to six, two F1s, two ICs and now two specials. The Formula 1 Register developed the notation 'P1', 'P2', to denote extra cars built by the Parnell team and their notation is used for "VR" ['P1'] and the extra standard car ['P2'].
As Ray so wisely says, "The things we don't know."