Many sources of uncertainties limit the accuracy and precision of climate projections. Here, we focus on the parameter uncertainty, i.e. the imperfect knowledge of the values of many physical parameters in a climate model. We use LOVECLIM, a global three-dimensional Earth system model of intermediate complexity and vary several parameters within their range of uncertainty. Nine climatic parameter sets and three carbon cycle parameter sets are identified. They all yield present climate simulations coherent with observations and they cover a wide range of climate responses to doubled atmospheric CO2 concentration and freshwater flux in the North Atlantic sensitivity experiments. They also simulate a large range of atmospheric CO2 concentrations in response to prescribed emissions. Climate simulations of the last millennium are performed with the 27 combinations of these parameter sets. A special attention is given to the ability of LOVECLIM to reproduce the evolution of several climate variables over the last few decades, for which observations are available. The model response, even its ocean component, is strongly dominated by the model sensitivity to an increase in atmospheric CO2 concentration but much slightly by its sensitivity to freshwater flux in the North Atlantic. The whole set of parameter sets leads to a wide range of simulated climates. Although only some parameter sets yield simulations that reproduce the observed key variables of the climate system over the last decades, all of them could be used to characterise extreme climate projections.