The Seebeck coefficient for a metal/metal joint is the temperature-dependence
of the interface voltage that reaches thermal equilibrium between two
materials, one with higher free charge density than the other. Just as
fur and amber result in charge separation when you rub them, so do
conductors swap a little bit of charge on contact. The usual kind of
electric circuit, with iron wire and copper wire joined in a loop, has
no net current resulting from the voltage difference, as long as the
two joints are at a common temperature. When, however, you
disturb that thermal equilibrium, the voltage at one junction no longer
balances that at the other...
OK, now that Seebeck coefficient might be 100 uV/C, a tiny fraction of
a volt per Celsius degree. With boiling water and ice, you'd get
about 1/100 of a volt total. The effect is reversible, and applying
1/100 of a volt would cause charges to flow, and (like gas being compressed)
at one junction the flow causes heat to be generated, while (like
gas being expanded) the flow at the other junction causes heat to be
absorbed. That's Peltier cooling.
The 'thermally in parallel' claim tells us that a number of dissimilar
materials are all connected thermally to a single HOT reservoir at the
P-to-N junction, and all connected thermally to a single COLD reservoir
at the N-to-P junction.
The 'electrically in series' claim tells us that those junctions are
all in series (if there's 100 pairs of them, with 1/100 volt each, you get 1 volt
for the device).
It is done in series electrically so drive voltages aren't impractically
low for application of significant power, and it is done in parallel
thermally so that all the junction pairs pump equal amounts of heat. That
makes a (possibly small) temperature difference, but a maximally large heat
flow from the apparatus. If all the heat-flow elements WEREN'T thermally
in parallel, the cooling would apply somewhere other than your target reservoir.
Now, as to the reason for using semiconductors: the important character of
semiconductors is that they can be doped to conduct electricity (and thus
complete the electrical circuit that powers the heat pump), AND they can
be doped so that the N type has lots of electrons (high concentration of
carriers) while the P type has very few N carriers. That maximizes
the Seebeck coefficient because it makes a highly asymmetric electron
population. Instead of metals and maybe 1 volt power, you can use
semiconductors and 10 volt power.
Choosing the 'right' semiconductors also involves mechanical bonding,
thermal conductivity (which 'leaks' heat and reduces efficiency), and
electrical resistance (which creates waste heat). The electrical bonding
is also important, the junctions must carry current in TWO directions,
and that requires some ohmic-contact materials wizardry.