To improve coding efficiency by exploiting the local inter-component redundancy between the luma and chroma components, the cross-component linear model (CCLM) is included in the versatile video coding (VVC) standard. In the CCLM mode, linear model parameters are derived from the neighboring luma and chroma samples of the current block. Furthermore, chroma samples are predicted by the reconstructed samples in the collocated luma block with the derived parameters. However, as the CCLM design in the VVC test model (VTM)-6.0 has many conditional branches in its processes to use only available neighboring samples, the CCLM implementation in parallel processing is limited. To address this implementation issue, this paper proposes including the neighboring sample generation as the first process of the CCLM, so as to simplify the succeeding CCLM processes. As unavailable neighboring samples are replaced with the adjacent available samples by the proposed CCLM, the neighboring sample availability checks can be removed. This results in simplified downsampling filter shapes for the luma sample. Therefore, the proposed CCLM can be efficiently implemented by employing parallel processing in both hardware and software implementations, owing to the removal of the neighboring sample availability checks and the simplification of the luma downsampling filters. The experimental results demonstrate that the proposed CCLM reduces the decoding runtime complexity of the CCLM mode, with negligible impact on the Bjøntegaard delta (BD)-rate.