I do not believe the same font is being used in both pictures

Both fonts are "Times".

Both fonts are "Times".

The parentheses are different.

Are you using LuaTeX/XeTeX and the unicode -math package?

I used pdfLaTeX, here is a link to the overleaf https://www.overleaf.com/read/sskgsdhqvsbb. Here is how it looks with unicode-math and LuaLaTeX:

This is really asking for #360, IMO. There's nothing wrong with how it is being laid out in Typst, it's the size of the parentheses that is not the same as in LuaTeX. For example, if I scale the parens in Typst down and in LuaTeX up, the subscript positioning is the same.

Normal	Debug

\documentclass[11pt]{article}
\usepackage[paperheight=2cm,paperwidth=4cm,margin=0cm]{geometry}
\usepackage{fontspec}
\usepackage{unicode-math}
\usepackage{lua-visual-debug}
\setmathfont{New Computer Modern Math}
\begin{document}

\[ \left(\frac{d \mu}{d t}\right)_p
 \biggl(\frac{d \mu}{d t}\biggr)_p \]

\end{document}

#set page(width: 4cm, height: 2cm)
#set align(bottom)

$
  lr(((d mu)/(d t)), size: #90%)_p
  ((d mu)/(d t))_p
$