I agree, the audio in the lesson sounds incorrect; however, while played in the discussion page it sounds correct.
What I found from Wiktionary is that the pronunciation is "pɐˈkazɨvəjʊ". In non-IPA terms, it is pronounced exactly as it is spelled, with the stress located on the first "а" and the ending as "a-yu".
Покажу is first-person, perfective, future. "I will show." Показываю is first-person, imperfective, present. "I show/am showing."
(All present-tense forms of a verb are always imperfective, since it is assumed that if you state it in the present tense, you are still in the process of performing the action. True perfectives, such as показать, whence cometh покажу, will not have a present tense form of the verb. For your consideration: https://en.openrussian.org/ru/показать, and note the conjugation tables which lack a present tense. Then compare to показывать at the same site. Perfective connotes that the action was successfully completed or a one-off activity, while imperfective connotes that the action was unfinished, a habitual/repeated event, done over a length of time, or references one's state (such as, I was at home).)
Differentiation between perfective/imperfective adapted from http://www.russianlessons.net/grammar/verbs_aspect.php.
Все is nominative; a literal translation of the Russian would be "I show the park to everyone" (or a similar sentence depending on the context), so dative всем is needed to express this "to everyone".
OK . this is gramatically correct. but then why is PARK not in the accusartive form ? gramatically it IS an accusative because it answers to the question " I SHOW WHAT TO WHOM " or ' TO WHOM DO I SHOW WHAT" shouldn't it be PARKU ? or does Russian say THE PARK IS SHOWN TO EVERYBODY ? but then the verb should be different ?
Парк is in accusative. The accusative form of inanimate nouns is the same as nominative.
Only in Accusative case. There is a distinction between animate/inanimate in masculine singular and in masculine and femenine plural. And inanimates keep the Nominative form, so it's easy ;)
See the unit "Accusative case", i think it's explained there (in web version).