There are twelve different numbers in the list, so logically it can’t be done in fewer than 7 parts (3 of one hand and four of the other, to combine in 3 x 4 = 12 different ways).

However the fact that 23 is the only odd number in the list throws a spanner in the works as it effectively means one part will help make up the 23 but not be used elsewhere. Assuming that you can find a way of making the other 11 with just 3 + 4 distinct parts, 8 parts in total will be required.

If we assume for the moment that there are only 3 L parts, and that one of them has length 0 (impossible ultimately, but useful right now), then what we are searching for is a subset of four of the 11 lengths (so disregarding 23), and two number a and b such that the remaining 7 numbers can be made by adding either a or b to one of the 4 numbers.

By experimenting, the subset 20, 44, 48 and 50, with a and b values 8 and 26 fits the bill. To avoid the zero length part, we can simply subtract 10 from each of (20,44,48,50) and call those the L parts, and add 10 to (0,8,26) to find the R parts, adding in an extra R part to make 23 possible. Thus:

20 = 10L + 10R

23 = 10L + 13R

28 = 10L + 18R

44 = 34L + 10R

46 = 10L + 36R

48 = 38L + 10R

50 = 40L + 10R

52 = 34L + 18R

56 = 38L + 18R

58 = 40L + 18R

70 = 34L + 36R

74 = 38L + 36R

which uses the following eight parts: 10L, 34L, 38L, 40L, 10R, 13R, 18R, 36R.