Hi,
There will be and should be a big difference between the connections where all are in series and all are in parallel. The short answer is that in the parallel connection all the gains are added and that even includes those filters sections that are inactive, while in the series connection the gains are multiplied so any sections with gain 1 just pass along the signal to the next stage. We can look at this in more detail.
First, a single filter will pass signals with different gains at different frequencies, and we look at least at two different gains. The first gain would be the pass band gain, and the second would be the stop band gain. We can call these Gp and Gs respectively. This means for three filters we have three sets of gains to consider as a minimum:
Gp1, Gs1, and Gp2, Gs2, and Gp3, Gs3. The numbers 1 to 3 indicate which of the three we are talking about.
The total gain G can now be computed by knowing that only one filter is operating in it's pass band at a time. It does not matter if they overlap right now because we only look at the two frequencies for each, at least for now. For a single tone test frequency this gives total series gain GS and total parallel gain GP as:
GS=Gs1*Gs2*Gs3
GP=Gs1+Gs2+Gs3
We can see there is a world of difference already because in one config the gains are multiplied, and in the other config they are added. We have one more gain to consider, and that is the full response gain which we can call G4, and that gets added to the others in BOTH configs:
GS=Gs1*Gs2*Gs3+G4
GP=Gs1+Gs2+Gs3+G4
So you see we have a problem. The stop band gains in the first config above would have to be all 1 (one), while in the second config the stop band gains would have to be all 0 (zero). These numbers refer to the gain Vout/Vin and not the gain in db.
For example, if we had all stop band gains equal to 1 instead of 0 and we had one filter in the pass band with gain of 2, and G4=1/2, the two configs would yield:
GS=1*1*2+0.5=2.5
GP=1+1+2+0.5=4.5
You can see there is an almost 2:1 difference in total gain at that one frequency, and we only have to show that is does not work for one frequency to prove that it wont work for a given single frequency. It's also true that if we make all the stop band gains zero then the two gains still come out different. To be sure we'd have to look at two test frequencies, but you can see where this is going.
The conclusion then is what works for one config does not work for the other.
If you are getting the RIGHT response from the series connections, then you have to keep it that way without altering every filter. There is also a chance that the filters can not be easily altered in a way that would allow the other config.
Lucky these circuits are not that hard to analyze analytically either so we could work up a total analytical solution if need be. If the simulations work for you then that's good too and saves some work.