Yes you can increase sensitivity in the limit case (but probably not as usefully as you are hoping), so here goes:
Ultimately, sensitivity is limited by signal/noise ratio.
The ultimate noise limit is set by thermal noise, and at 20C it is a fixed power (-174dBm/Hz)
The Signal voltage is fixed by bridge voltage * bridge gain.
Signal power is set by V^2/R.
So if you parallel two bridges (both under strain), Signal voltage remains constant, R=1/2, and Power=2x.
So SNR improves by 2x or 3dB.
And sensitivity improves by sqrt(2).
(You can of course just use a single lower resistance bridge element)
Now for this to be useful, your sensitivity has to be limited by thermal noise.
For bridge circuits, it is more often limited by thermal drift, hysteresis, and mechanical processes that produce analogous effects to 1/f and 1/f^2 electronic noise.
One reason for this is that bridges are often used in very low bandwidth applications. If you are weighing a truck, the bandwidth is less than 1Hz, so the thermal electrical noise is very small.
Now this would change if you tried to use your bridge at very high speed. e.g as an ultrasonic microphone. If the signal bandwith is large, and >10kHz, then thermal electrical noise may well become the limiting factor, and reducing bridge R would help.
As I said, mechanical noise may be the limit not electrical noise. In this case, two separated sensors might get uncorrelated mechanical noise, (but correlated signal), and in this case you will also improve sensitivity.
Also while normally you would choose to use a single lower resistance bridge to improve sensitivity, this has a thermal limit as the bridge power dissipation is increasing as you lower R. At some point, you have to stop, and then using multiple separate bridges becomes your only option to further reduce R.
AC bridge amplifiers using microphone transformers would let you keep improving SNR down to less that 1 ohm of resistance - a 1000x improvement from a 1k bridge!
Also realise that you can combine signals after the amplifiers. This will equally reduce noise, but also reduces the amplifier noise as well. (it also opens up options for beamforming to further improve SNR when signals or interference comes from a particular direction, even in non-obvious areas like temperature sensing)