How to use blackboard bold numbers with the Palatino (mathpazo) font?

\documentclass{article}
\usepackage{bbold}
\let\altmathbb\mathbb
\usepackage[sc]{mathpazo}

\begin{document}    
Hello world $\altmathbb{012345}$    
\end{document}

To restore the original so that no syntax variant is required:

\documentclass{article}
\usepackage{bbold}
\let\altmathbb\mathbb
\usepackage[sc]{mathpazo}
\AtBeginDocument{\let\mathbb\altmathbb}
\begin{document}    
Hello world $\mathbb{012345}$  
\end{document}

There are several options.

The mathalpha Package

This package contains both a pazo and a px blackboard bold alphabet, matching mathpazo.

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{mathpazo}
\usepackage[bb=pazo]{mathalpha}

\begin{document}
\( ABC \mathbb{ABC} \)
\end{document}

You could also replace the mathalpha option bb=pazo with bb=px. The latter comes with a bold double-struck alphabet usable as \mathbbb. Since mathpazo is a fork of Adobe’s version of Palatino, any font package matching Palatino matches it. See the mathalpha package documentation for the complete list of alphabets.

The newpxmath Package

This is essentially an upgrade to mathpazo. It still uses legacy 8-bit fonts, instead of the more modern Unicode and OpenType, but it’s compatible with PDFLaTeX. It includes a much broader repertoire of symbols, including several double-struck alphabets.

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{newpxtext, newpxmath}
\usepackage{parskip}

\begin{document}
\( ABC \mathbb{ABC} \\
   \varmathbb{ABC} \)
\end{document}

This comes with three different blackboard-bold alphabets, which are controlled by package options to newpxmath. If you like the letters on the second line, you can make those the default with \usepackage[varbb]{newpxmath}, or you can substitute math alphabets with mathalpha.

Since newpx, like mathpazo, is a clone of Palatino, you can match it with any other Palatino clone (such as tgpagella) or a font matching Palatino (such as Hermann Zapf’s other masterpieces, Optima and Euler).

The Modern Toolchain

If you are able to use unicode-math and a modern TeX engine, I personally recommend it. Asana Math is a modern, Unicode math font based on mathpazo. TeX Gyre Pagella Math is also similar.

\documentclass{article}
\usepackage{unicode-math}
\usepackage{parskip}

\defaultfontfeatures{Scale=MatchLowercase}
\setmainfont{TeX Gyre Pagella}[Scale = 1.0]
\setmathfont{Asana Math}

\begin{document}
\( ABCabc \\
   \mathbb{ABCabc}
\)
\end{document}

\documentclass{article}
\usepackage{unicode-math}
\usepackage{parskip}

\defaultfontfeatures{Scale=MatchLowercase}
\setmainfont{TeX Gyre Pagella}[Scale = 1.0]
\setmathfont{TeX Gyre Pagella Math}

\begin{document}
\( ABCabc \\
   \mathbb{ABCabc}
\)
\end{document}

If you wanted to use just the blackboard-bold alphabets of TeX Gyre Pagella Math, and other symbols from Asana Math, you can do that with

\setmathfont{Asana Math}
\setmathfont[range={bb,bbit}]{TeX Gyre Pagella Math}

Or vice versa. If you have an authentic copy of Palatino on your system, you can also use that as your text font instead of the clone, Pagella.