Context. Observations of molecular gas have played a key role in developing the current understanding of the late stages of stellar evolution. Aims: The survey Planetary nebulae AND their cO Reservoir with APEX (PANDORA) was designed to study the circumstellar shells of evolved stars with the aim to estimate their physical parameters. Methods: Millimetre carbon monoxide (CO) emission is the most useful probe of the warm molecular component ejected by low- to intermediate-mass stars. CO is the second-most abundant molecule in the Universe, and the millimetre transitions are easily excited, thus making it particularly useful to study the mass, structure, and kinematics of the molecular gas. We present a large survey of the CO (J = 3-2) line using the Atacama Pathfinder EXperiment (APEX) telescope in a sample of 93 proto-planetary nebulae and planetary nebulae. Results: CO (J = 3-2) was detected in 21 of the 93 objects. Only two objects (IRC+10216 and PN M2-9) had previous CO (J = 3-2) detections, therefore we present the first detection of CO (J = 3-2) in the following 19 objects: Frosty Leo, HD 101584, IRAS 19475+3119, PN M1-11, V* V852 Cen, IC 4406, Hen 2-113, Hen 2-133, PN Fg 3, PN Cn 3-1, PN M2-43, PN M1-63, PN M1-65, BD+30 3639, Hen 2-447, Hen 2-459, PN M3-35, NGC 3132, and NGC 6326. Conclusions: CO (J = 3-2) was detected in all 4 observed pPNe (100%), 15 of the 75 PNe (20%), one of the 4 wide binaries (25%), and in 1 of the 10 close binaries (10%). Using the CO (J = 3-2) line, we estimated the column density and mass of each source. The H2 column density ranges from 1.7 × 1018 to 4.2 × 1021 cm-2 and the molecular mass ranges from 2.7 × 10-4 to 1.7 × 10-1 M⊙. The reduced spectra are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/618/A91