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Consider the following MWE:

documentclassarticle
usepackage[sc]mathpazo
usepackagebbold

begindocument 
Hello world $mathbb012345$ 
enddocument

This gives me

While removing the mathpazo package I get

How can I have it both ways? I would like to use the Palatino font, but at the same time use blackboard-bold numbers.

There are several options.

The mathalpha Package

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

documentclassarticle
usepackage[T1]fontenc
usepackagemathpazo
usepackage[bb=pazo]mathalpha

begindocument
( ABC mathbbABC )
enddocument

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 package documentation for the complete list of options.

The newpxmath Package

This is essentially a more modern replacement for mathpazo. It includes a much broader repertoire of symbols, including several double-struck alphabets.

documentclassarticle
usepackage[T1]fontenc
usepackagenewpxtext, newpxmath
usepackageparskip

begindocument
( ABC mathbbABC \
 varmathbbABC )
enddocument

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.

documentclassarticle
usepackageunicode-math
usepackageparskip

defaultfontfeaturesScale=MatchLowercase
setmainfontTeX Gyre Pagella[Scale = 1.0]
setmathfontAsana Math

begindocument
( ABCabc \
 mathbbABCabc
)
enddocument

documentclassarticle
usepackageunicode-math
usepackageparskip

defaultfontfeaturesScale=MatchLowercase
setmainfontTeX Gyre Pagella[Scale = 1.0]
setmathfontTeX Gyre Pagella Math

begindocument
( ABCabc \
 mathbbABCabc
)
enddocument

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

setmathfontAsana 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.

documentclassarticle
usepackagebbold
letaltmathbbmathbb
usepackage[sc]mathpazo

begindocument 
Hello world $altmathbb012345$ 
enddocument

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

documentclassarticle
usepackagebbold
letaltmathbbmathbb
usepackage[sc]mathpazo
AtBeginDocumentletmathbbaltmathbb
begindocument 
Hello world $mathbb012345$ 
enddocument