They both would benefit from some more evocative English.
18 θεὸν οὐδεὶς ἑώρακεν πώποτε· μονογενὴς θεὸς ὁ ὢν εἰς τὸν κόλπον τοῦ πατρὸς ἐκεῖνος ἐξηγήσατο.
18 nobody has seen God yet; the only child God (who is in the lap of the father) that one led (the knowledge of him) out.
Italics are my additions.
"Who is at the Father's side" is a pretty weak translation in my opinion. See https://logeion.uchicago.edu/κόλπος for a good sense of how "κόλπος" specifically invokes one's "bosom" or lap, e.g. putting something inside your clothes to carry it around, or even being in the womb, and metaphorically then extended to other kinds of cavities. It implies a much closer and intimate relationship than "beside".
And yes, a translation of verse 14 probably should include "coming from the side of something" since that is the basic cognitive model of παρὰ with a genitive object (rather than "at the side of" for dative or "to (or past) the side of" for accusative):
14 Καὶ ὁ λόγος σὰρξ ἐγένετο καὶ ἐσκήνωσεν ἐν ἡμῖν, καὶ ἐθεασάμεθα τὴν δόξαν αὐτοῦ, δόξαν ὡς μονογενοῦς παρὰ πατρός, πλήρης χάριτος καὶ ἀληθείας·
14 and the word became flesh and pitched his tent among us, and we observed his reputation, reputation as only from the side of the father, full of favor and truth;
The meanings of prepositions and cases are a bit more abstract than a meaty noun like κόλπος, but by the same token they map less well from Greek to English, and are more frequently translated to overly-simple glosses.
I wouldn't assume in every case that one English verse affects how another should be translated, but in this case the Greek bears out for both verses.