Early Buddhism holds that there are six consciousnesses, 5 that occur in conjunction with the senses, and 1 'idea sense'. In the Yogacara philosophy which is taken up by most Mahayana school (Chinese+Tibetan), this was developed into https://en.m.wikipedia.org/wiki/Eight_Consciousnesses So, 'extra sensory' in the Buddhist context, is still perception, with the mind, in various ways, including verifying the law of karma. In modern framing we understand the Earth having a liquid magnetic core, cyanobacteria producing oxygen, and our own genetic lineages. When you look at the actual Buddhist account of karma which sets it in vast space and time (just 1 kalpa is around 3 x the current age of the universe), rather than tge popular idea which is like the Hindu description, it is quite comparable to that modern thinking.
There is an issue like with all pre-scientific theology, in distinguishing between metaphor and literalism, as we use those categories now. A strong case can be made for Buddha presenting thought experiments (parable of the arrow, the mother and the mustard seed), and reinterpreting ideas of his time to serve philosophical and ethical insights (karma, cosmology, conditionality); but presenting these in modern understanding as literally true. Voyaging into other realms sounds 'extra sensory', but again when you look at the actual texts https://www.accesstoinsight.org/lib/authors/jootla/wheel414.html it sounds more like theology, dealing with Brahma as 'causeless cause', and with dismissing appeals to external beings that distract us from our own psycology and choices.
'Extra sensory perception' is poorly defined in Western thought too. Are ideas, like infinity, 'extra sensory'? You have to say yes, but it's not what people jump to. Birds and cattle sense the Earth's magnetic field, and humans have the same type of cells they do it with. 'Extra-sensory'? Or just another sense? Modern reviews list us having 42 senses, from our stomachs acidity to proprioception. Any 'new' way of sensing that was demonstrated to actually work, would be added to the list. While, we 'voyage' into all kinds of othervrealms in modern physics, like mathematical phase spaces where we arrange information to identify manifolds or surfaces with specific properties, and draw out results of importance for 'ordinary space'. But even that latter, is fractal, almost certainly holographic, and far from our simple intuition of 'material' or even 'real'.
Plato's story of Er https://en.m.wikipedia.org/wiki/Myth_of_Er is remarkably like Buddhist thinking, such as in The Tibetan Book Of The Dead. I just don't accept the distinction between Eastern & Western thinking on ESP. Only, between ancient and modern, and even that mainly as an issue of terminology. In our modern thinking, ESP doesn't really make sense as a category, and we might draw inspiration in fact from descriptions of realms in Buddhist thought, as from maths and physics, in countering our intuitions and examining the actual nature of what is 'reality'.