How to 'Shift in Parallel' in a shift register?
You just shift 4 times.

Early calculators used a 4 bit CPU, as that is all you need for BCD maths.

But inside that 4 bit cpu, they used a 1 bit arithmetic-logic unit to do the actual adds. They shifted the 4 bits serially through the 1 bit ALU, doing 1 bit add-with-carry. (later 8 bit cpus did the same)

Once you get your head around this idea, you realise that doing multiple precision maths is no harder on a 1 bit ALU, that it is on a 32bit CPU - just slower, but using way fewer transistors. 

It is still multiple precision maths. 

And if you are making a calculator you will still have to do multiple precision maths even if you use a 32 bit ARM.

It seems to me, that you only need two, 16 bit one way shift registers (operands X,Y), with one 4 bit parallel load input nibble.

You can copy X to Y by 16 single bit shifts.
You can shift left by 15 rotate rights
You can do maths with a 1 bit ALU with carry.
You will need a clever decimal adjust every nibble.
You need a loadable 4 bit counter for shift control as you will have 4 bit, 1 bit,15bit and 16 bit shifts

So I really don't think you need bi-directional or parallel load shift resisters at all. 
Most of the logic operations and muxing  will all happen 1 bit wide at the end of the SR.