Steam carrot slices over boiling water for 10 to 12 minutes until very soft. Mash, spread on a plate to release steam and cool for 10 minutes; blot away any visible moisture.
Divide cream cheese into three equal portions. Process one portion with the beet, 1/2 teaspoon lemon juice and a pinch from the measured salt until smooth, then scrape into a bowl.
Clean the processor and blend a second portion with the cooled carrot, remaining lemon juice and another pinch of salt. Clean again and blend the last portion with parsley, remaining salt and black pepper.
Build four identical stacks, each using a white slice, one-quarter of the beet filling, a wheat slice, one-quarter of the carrot filling, a white slice, one-quarter of the parsley filling and a final wheat slice. Spread each filling evenly to the corners.
Wrap the four stacks snugly and refrigerate for at least 60 minutes. Keep them flat without a heavy weight, which would squeeze the colored fillings out.
Trim the crusts with a sharp serrated knife. Cut each stack into three equal strips, then halve each strip crosswise to make six small rectangles; yield 24 ribbon sandwiches, two per serving.
Keep covered and refrigerated until serving within 24 hours, separating layers in the container with parchment so the cut faces stay clean.