Er, the parsing is done ahead of time. The BASIC interpreters then were byte-code interpreters; the one of the beeb a particular fast one.

I know nothing about the first ARM, but ARM2 of Archimedes (anno 1987) was significantly faster than a MC68k (both at 8MHz), both much faster than a 6502 at one (typical) or two (in the beeb) MHz.

A BASIC interpreter using the ARM 1 or 2 might not have been literally faster than machine code on a 6502 (certainly not for some silly micro benchmarks), but, the stated goal, allowing high level programming where earlier assembly was required, certainly was met.

The early 8MHz ARM2-based Archimedes machines arguably also outperformed contemporary 16/20MHz 80386 machines (due to the x86-based machines being slower to access RAM before the advent of on-motherboard cache and zero waitstate RAM) as well.