Could it be possible/reasonable to extract Peltier thermoelectric energy by using a long rod of suitable material that connects the cold bottom of the ocean and the warm surface layer?
It's theoretically possible, and working prototypes have been built on an experimental basis to explore the possibilities, but to the best of my knowledge no large scale, semi- commercial projects have yet been undertaken. I say semi-commercial because it is likely that some form of state subsidy would be necessary, at least in the early stages.
Not really. If you use highly thermally conductive rods (like copper or aluminium), the depth of the ocean (let's say is about the order of 5 km) is so much greater than the radius of the metallic rod that radial heat conduction will dominate over the longitudinal one. It means that you won't be able to carry the "coldness" (or "hotteness" for that matter) of deep underwater temperatures to the surface. What will happen instead is that the temperature of the rod will be roughly the same as the temperature of the water around it. So, at the surface where your Peltier element (or better to say, thermoelectric generator) there is virtually no temperature difference due to the rod (or heat exchanger). The rod will be essentially entirely useless. And this remains true even if you somehow encapsulate the rod in a vacuum cylinder so that the rod makes no contact with water. In that case the thermal resistance of the rod (which scales linearly with its length) is so big that the difference in temperature between the top of the rod and the bottom of the rod will be vanishingly small compared to the temperature difference between the surface of the ocean and its bottom.
In other words, no matter what you do with the rod, using a rod between the surface of the ocean and its bottom in order to establish a temperature difference across a thermoelectric generator will not work as you expect. The rod will not be able to transfer the heat from the bottom to the top, or viceversa. An analogy with an electrical circuit would be that you have in mind that using a copper wire between two elements will be fine because it conducts electricity well. However the reality is that the wire is such that its resistance is insanely high despite that its conductivity is also very high. For example if the wire has a diameter of a few micrometers and a length of millions of kilometers, you end up having a resistance of mega ohms despite the fact that you used copper. That's roughly equivalent to the case you have in mind.
It's possible to use the temperature difference between ocean surface water and deep water to extract energy from the ocean, but the thermodynamic efficiency of the process is so low that the capital expense cannot be paid for in a reasonable time frame. Thus, from an economic standpoint, this process isn't reasonable. For an example that is closely related to your description, see https://en.wikipedia.org/wiki/Ocean_thermal_energy_conversion