The first part of the question is comparing two completely different pipe lines of different diameter. The small pipe diameter needs more pressure to move the same flow rate. Bernoulli principle applies to points within the same pipe line.
When both pipes are within the same pipeline pressure is lower in the small diameter pipe, but compared to what? Compared to the larger pipe upstream, the pressure will be lower. Why? Because energy (in this case pressure, or potential energy, from an upstream point must be used to accelerate the fluid as it moves into the smaller area of a smaller diameter pipe. So the pressure there in the smaller diameter pipe will be lower than the upstream point, but velocity will also be higher. Potential energy was converted to kinetic energy, which has increased just as much as its potential energy (pressure) has decreased.
Should the next part of the pipeline get large again, then the small pipe's kinetic energy from its high velocity will be converted to higher pressure as the velocity slows down and enters the large diameter.
To make this all right and proper, you must think in terms of total energy, which is always the same in pipes of both large and small diameters, as long as they carry the same flow rate and that there will be variations of how much of that energy is kinetic and how much will be potential energy when you have changes of diameters in the same pipeline.