The rules for capitalisations of name prefixes like 'van', 'von', ... wary wildly between languages and regions. Since you load babel with the dutch option it is fair to assume you either want the Dutch spoken in the Netherlands or Flanders.
It is my understanding that in the Netherlands Dutch tussenvoegsels like 'de', 'van' and 'van der' are capitalised unless they are preceded by the first name (or name initial). To quote the slightly confusing English Wikipedia page on the subject Tussenvoegsel
According to Dutch language rules in the Netherlands, the tussenvoegsel in a surname is written with a capital letter only when it starts a sentence or is not preceded by a first name or initial.
(If the tussenvoegsel starts the sentence then it is not directly preceded by a first name or initial anyway...)
Example from Dutch media outlets: 'vertelde historicus Maarten van Rossem' vs. 'historici hadden volgens Van Rossem' https://www.volkskrant.nl/es-b7bd1412 or 'presentatiecollega Maarten van Rossem' https://www.groene.nl/artikel/o-la-la vs. 'In Nederland Van Rossem, Tromp, Koen Koch, Von der Dunk, ...' https://www.groene.nl/artikel/wie-is-schuldig.
Cf. also https://woordenlijst.org/leidraad/16/2, https://taaladvies.net/taal/advies/vraag/433, https://onzetaal.nl/taaladvies/hoofdletters-in-namen-nynke-van-der-sluis-nynke-van-der-sluis/ and the beautiful pictures in https://zichtbaarnederlands.nl/zn/spelling-achternamen.php
In Belgium (Flanders) the capitalisation of the tussenvoegsel depends on the 'official form' of the name as on official documents like IDs or passports. It is usually stated that names do not change capitalisation All examples I found assumed that at least the first tussenvoegsel is capitalised at all times. It would be interesting to see how names with a lowercase v would be treated without first name and at the beginning of a sentence...
From this I understand that Belgian tussenvoegsels should be input with capital letters in the .bib file as well, i.e. author = {Van Rossem, Jean-Pierre} (this will lead Biber/BibTeX to treat the Van as part of the family name and not as a prefix, which nicely fits with all sorting rules and the interpretation of https://zichtbaarnederlands.nl/zn/spelling-achternamen.php).
Given all these rules it seems to me that it makes little sense to use the capitalised \Cite form at all times. Instead I would change the name formatting macros to apply capitalisation if needed. The following implements the rules w.r.t capitalisation used in the Netherlands as I understand them
\documentclass{article}
\usepackage[dutch]{babel}
\usepackage[autostyle]{csquotes}
\usepackage[style=ext-authoryear, giveninits=true, maxnames=3, doi=false, isbn=false, dashed=false, natbib=true, backend=biber]{biblatex}
\DeclareDelimFormat{nameyeardelim}{\addspace}
\SetCiteCommand{\parencite}
\newbibmacro*{name:family}[4]{%
\ifuseprefix
{\usebibmacro{name:delim}{#3#1}%
\usebibmacro{name:hook}{#3#1}%
\ifdefvoid{#3}
{}
{\mkbibnameprefix{\MakeCapital{#3}}\isdot
\ifprefchar{}{\bibnamedelimc}}}
{\usebibmacro{name:delim}{#1}%
\usebibmacro{name:hook}{#1}}%
\mkbibnamefamily{#1}\isdot}%
\newbibmacro*{name:given-family}[4]{%
\usebibmacro{name:delim}{#2#3#1}%
\usebibmacro{name:hook}{#2#3#1}%
\ifdefvoid{#2}{}{\mkbibnamegiven{#2}\isdot\bibnamedelimd}%
\ifdefvoid{#3}{}{%
\ifdefvoid{#2}
{\mkbibnameprefix{\MakeCapital{#3}}\isdot}
{\mkbibnameprefix{#3}\isdot}%
\ifprefchar
{}
{\ifuseprefix{\bibnamedelimc}{\bibnamedelimd}}}%
\mkbibnamefamily{#1}\isdot
\ifdefvoid{#4}{}{\bibnamedelimd\mkbibnamesuffix{#4}\isdot}}
\newbibmacro*{name:family-given}[4]{%
\ifuseprefix
{\usebibmacro{name:delim}{#3#1}%
\usebibmacro{name:hook}{#3#1}%
\ifdefvoid{#3}{}{%
\mkbibnameprefix{\MakeCapital{#3}}\isdot
\ifprefchar{}{\bibnamedelimc}}%
\mkbibnamefamily{#1}\isdot
\ifdefvoid{#4}{}{\bibnamedelimd\mkbibnamesuffix{#4}\isdot}%
\ifdefvoid{#2}{}{\revsdnamepunct\bibnamedelimd\mkbibnamegiven{#2}\isdot}}
{\usebibmacro{name:delim}{#1}%
\usebibmacro{name:hook}{#1}%
\mkbibnamefamily{#1}\isdot
\ifdefvoid{#4}{}{\bibnamedelimd\mkbibnamesuffix{#4}\isdot}%
\ifboolexpe{%
test {\ifdefvoid{#2}}
and
test {\ifdefvoid{#3}}}
{}
{\revsdnamepunct}%
\ifdefvoid{#2}{}{\bibnamedelimd\mkbibnamegiven{#2}\isdot}%
\ifdefvoid{#3}{}{\bibnamedelimd\mkbibnameprefix{#3}\isdot}}}
\usepackage{filecontents}
\begin{filecontents*}{\jobname.bib}
@book{vanOostrom2013,
langid = {dutch},
location = {Amsterdam},
title = {Wereld in Woorden},
publisher = {Uitgeverij Bert Bakker},
date = {2013},
author = {van Oostrom, Fritz},
options = {useprefix=true}
}
\end{filecontents*}
\addbibresource{\jobname.bib}
\begin{document}
\begin{itemize}
\item \cite{vanOostrom2013} is not what we want. the V must always be capitalized.
\item luckily \cite{vanOostrom2013} will fix this for us
\item same for \citeauthor{vanOostrom2013}, \citet{vanOostrom2013} and even \autocite{vanOostrom2013}
\item but \fullcite{vanOostrom2013}
\end{itemize}
Sometimes, however, I want to quote \textcquote*{vanOostrom2013}{Some Famous words}.
\printbibliography
\end{document}

Note that the name order with inverted names in the bibliography does not seem to follow the conventions in the Netherlands (I would expect 'Oostrom, F. van', see https://taaladvies.net/taal/advies/vraag/145/). For that you would need How can I put a name's prefix in front in citations but *not* in the bibliography?. A related question is at Prefixes in author names in references and bibliography.
It is still advisable to use the capitalised form \Cite, \Parencite at the beginning of sentences since those commands may produce more text that might be subject to capitalisation (one real-world example is "ibid.").
\Citet,\Citealt, and\Citeauthorare there for. In contrast, there should be no need to uppercase "de" in the sentence "As argued by de Vries (1999), ...". – Mico Dec 14 '18 at 12:22"this is a \textcquote" (Van Oostrom:2013:52)and 2)something related and Van Oostrom agreed (2013:52). So yes: always :/ – LooselySubtle Dec 14 '18 at 12:41\citewith\Citeglobally:\SetCiteCommand{\Cite}. If you need more flexibility you will have more work to do. – Paul Stanley Dec 14 '18 at 12:43I changed
– LooselySubtle Dec 14 '18 at 12:47\SetCiteCommand{\parencite}to\Parenciteand that seems to do the trick. Thanks!\citetand\citep) would long ago have implemented this rule. The very fact that\Citetand\Citeauthoreven exist strongly suggests that they were created to handle exceptional cases, viz., citation call-outs at the beginning of a sentence. Clearly, though, you're stuck with an intransigent and dogmatic "promoter", and there's nothing you can do about it. Good luck. – Mico Dec 14 '18 at 12:51If the authors name starts with a lowercase (such as Fritz van Oostrom) you maintain the lower case only if you use his fullname (Fritz van Oostrom says...). In all other cases, you transform the last name to uppercase (and this is what Van Oostrom says), to avoid confusion in your text...Yeah, it's not my professor/promoter :( – LooselySubtle Dec 14 '18 at 12:58natbib=trueoption that lets you use the capitalised\Citecommands, those are available even with the defaultnatbib=false.natbibonly enablesnatbib-like commands such as\citep(fot\parencite) and\citet(for\textcite) as well as\citealt,\citealpand a few other commands. Instead ofbibstyle=ext-authoryear,citestyle=authoryearI would just usestyle=ext-authoryear. And for\renewcommand*{\nameyeardelim}{\space}I would prefer\DeclareDelimFormat{nameyeardelim}{\addspace}(but that should not be needed if you dropnatbib=true). – moewe Dec 14 '18 at 13:15