「シャワーを[浴]{あ}びたら[公園]{こうえん}に[行]{い}く。」
is completely correct and very natural despite what your friend said. 「たら」 is used that way all the time among us native speakers. The nuance of that sentence is that you know you will be going to the park pretty soon, but you just want to take a shower first.
Nuance-wise, it is not as if taking a shower were the important "condition" for you to go to the park. In other words, this is fairly different from saying "If A, then B." both in meaning and feeling.
「たら」 is only "fully conditional" both in name and form when used in sentences such as:
「[雨]{あめ}が[降]{ふ}ったら、[明日]{あす}のピクニックは[中止]{ちゅうし}になります。」(If it rains, tomorrow's picnic will be canceled.)
「そんなにお[菓子]{かし}ばかり[食]{た}べてたら、[太]{ふと}るわよ。」(You will get fat if you keep eating snacks like that.)
I am sure that you could "feel" the difference in the use of 「たら」 between those two sentences and the sentence in question about showering and going to the park.
If the sentence in question had been:
「シャワーを浴びたら、公園に行ってもいいわよ。」
it would have been a full-fledged conditional sentence because it would then have meant "You may go to the park only if you take a shower first." A mother might say that to her kid.
Regarding the other sentence in question,
「シャワーを浴びて公園に行く。」
is certainly grammatical and "okay", but it is not necessarily more natural than the first sentence using 「たら」.
To my Japanese ears at least, that is a very character-less sentence. It is close to memo-writing in feeling -- "Will do A and B.".
It would be considerably more natural for a native speaker to say:
「シャワーを浴びてから公園に行く。」
if one wanted to imply that the order of the two actions happening is of some importance.
The te-form in general is useful, but it would help Japanese-learners to produce more natural-sounding sentences if they learned to use 「~~てから」、「~~て、それから」 instead of just using 「~~て」 in describing the order of consecutive actions.