Atlas of Pidgin and Creole Language Structures Online

Palenquero essentially uses a two-way contrast ('this one here', 'this one there'), and occasionally a third contrast ('this one there' (further removed than 'that one').The third contrast uses the word aké < Spanish aquel, and I suspect that its use is mainly superstrate-driven (it is not very common). We thus have:
ese akí (close)
ese aí (there)
ese aké (further removed)

Also, and this is important given the high frequency of aki and aí (two adverbs meaning 'here' and 'there' (cp. Span. aquí/allí), the two major locative adverbs have built in demonstrative meaning. Witness:

akí kasa 'in the (here) house'
aí kasa 'in the (there) house'

In Spanish, the above would correspond to:

esta casa
esa casa

For example:

Akí kasa ta asé kaló.
'In this house it is hot.'

For samples and further discussion, see Schwegler & Green (2007: 302ff).
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Note that ese is by far the most common "distance demonstrative", while ete < Spanish este is far less frequent than might be expected. To date, no explanation has been offered as to what diachronic factors contributed to the preferential treatment of ese over ete.

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Note that the two major mechanisms by which distance contrasts are expressed can also be combined so that they co-occur, as in:

ese ombe akí 'this man'
ese ombe aí 'that man'.

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Two-way contrast